1.1. From the Publisher.

This monograph combines a thorough introduction to the mathematical foundations of n-body Schrödinger mechanics with numerous new results.

1.2. From the Introduction.

It is our purpose in this monograph to present a complete, rigorous mathematical treatment of two body quantum mechanics for a wider class of potentials than is normally treated in the literature. At the same time, we will review the theory of the "usual" Kato classes, although no attempt has been made to make this review exhaustive or complete. The scope of what we present is best delineated by stating the limits of this work: we take for granted the standard Hilbert space formalism, and our main goal is to prove forward dispersion relations from first principles. For example we do not assume the Lippman-Schwinger equation but prove it within the framework of time-dependent scattering theory.

Mathematical physics of the type we discuss here seems to me to be of interest to the wider physics community for a variety of reasons. (a) It can often clarify concepts considerably. ... (b) It can uncover some interesting subtleties. ... (c) It can often serve as a guide to other fields, most notably high energy physics. ... (d) Finally, on a certain philosophical level, it is of use to know exactly which "reasonableness" arguments are really true and which require additional conditions.

1.3. Review by: Thomas Spencer.
Mathematical Reviews MR0455975 (56 #14207).

This monograph presents the basic mathematical techniques used in the study of quantum mechanics. Most of the theorems are proved and assume only a background in functional analysis. The proofs of some of the more complicated theorems are either sketched or referred to in the text's extensive bibliography. The first two chapters discuss some technical aspects of Hamiltonians defined with Rollnik class potentials which are a wider class than is usually considered. The third chapter applies the analytic Fredholm theory to study bound states and gives estimates on the number of bound states. The fourth and fifth chapters are devoted to scattering theory. The time dependent approach with an analysis of the wave operators is followed by a discussion of asymptotic completeness and lkebe's eigenfunction expansion. The next chapter investigates the analytic structure of the S-matrix and dispersion relations. The book concludes with a study of the equations of N body systems (the Weinberg-Van Winter equations) and establishes Hunziker's theorem on the location of the essential spectrum for the corresponding Hamiltonians.

2.1. From the Publisher.

This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modern mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained, although there are occasional references to later volumes. We have included a few applications when we thought that they would provide motivation for the reader. Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, modern physics, and partial differential equations.

2.2. From the Preface.

This volume is the first of a three-volume series devoted to an exposition of functional analysis methods in modern mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained although there are occasional references to Volumes II and III. We have included a few applications when we thought that they would provide motivation for the reader. Volumes II and III describe various advanced topics in functional analysis and give numerous applications to modern physics. Throughout the three volumes we have included important applications of functional analysis techniques to classical physics and partial differential equations. The chapter titles following the table of contents give some idea of ​​the topics in Volumes II and III.

2.3. Review by: Raul R Chernoff.
Mathematical Reviews MR0493419 (58 #12429a).

The authors of this volume do for mathematical physics what "Courant-Hilbert" did for an earlier generation: to marshal and narrate a large body of mathematics, much of it recent; to illuminate its connections with physical problems; and to make it into an accessible tool for future research. The mathematics is functional analysis and the theory of operators in Hilbert space, especially unbounded self-adjoint operators, which play a central role in quantum mechanics. Future volumes are promised treating operator algebras, group representation theory, probabilistic methods, and their applications to quantum field theory and statistical mechanics.

Volume I, "Functional analysis", is a relatively standard introduction to the subject. The authors take care to mention some strategic examples and applications, but the really substantial physical applications are deferred to later volumes. Each chapter in this volume and its successors has a good set of exercises as well as a sheaf of notes and remarks, including historical comments and a guide to the mathematics and physics literature. For example, the notes to Chapter VIII include a brief but surprisingly complete survey of the mathematical foundations of quantum mechanics à la Mackey.

2.4. Review by William Farris.
Bulletin of the American Mathematical Society 2 (3) (1980), 522-530.

With a knowledge of the equations of physics and with sufficient mathematical insight, one should be able to derive the physical properties of the world we see around us. The program would be to classify the elementary particles and analyse the various forces between them. Their motions should then follow from the laws of quantum mechanics.

The difficulty, of course, is that the classification is not complete and the forces are not known. There is a traditional division of forces into strong, electromagnetic, weak, and gravitational interactions. The first three of these play a role on the submicroscopic level and are presumably described by quantum fields. The present is a period of intense speculation about the nature of the fields responsible for the strong and weak forces. There is already a detailed theory of the electromagnetic field, but even that is not on a completely rigorous footing. So a complete mathematical description of nature is for the future.

It is possible, however, to come rather close to this ideal even now. Consider a world consisting of electrons and nuclei. Ignore most of the quantum mechanical features of the electromagnetic field; in fact, direct attention to the part of the electric field given by Coulomb's inverse square law. Treat this system with quantum mechanics, in the nonrelativistic approximation in which there is no particle creation. Add a few refinements, such as spin and the exclusion principle. The result should be a fairly good description of our world. This model should describe most physical and chemical properties of materials. It should explain why tables are solid and fire is hot, why the sky is blue and grass is green. It should not explain nuclear energy, radioactivity, or why apples fall, since these involve the other forces.

The task remains of carrying out this derivation. Is there anything interesting to say before going to the computer? There had better be, if we want to be able to make much sense out of physics. The series Methods of modern mathematical physics is an ambitious attempt to survey recent progress toward a rigorous qualitative description of quantum mechanical motion. (The series also contains considerable material on related mathematical problems, but the central theme is quantum mechanics.) The first two volumes in the series dealt with some preliminary functional analysis and with the determination of the time evolution. Volumes III and IV are companion volumes dealing with scattering and spectral theory. They describe the different possible kinds of motions of the individual particles or atoms or molecules.

2.5. Review by: J A Goldstein.
The American Mathematical Monthly 80 (10) (1973), 1152-1153.

According to the publisher, this book is intended as a "textbook for a first year graduate level course in functional analysis" and as a "reference text for the physicist interested in modern mathematical techniques of relevance to wide areas of physics." I used this book as a text for a one-semester course in functional analysis for first-year graduate students in mathematics who had completed a one-semester course in measure theory and integration. The lectures did not follow the text, but the students were required to read much of the text and work many of the exercises.

The students were critical of the book. They found it hard to read; they wished that Reed and Simon were less stingy with details. Also, they objected to occasional lapses into imprecision on the part of the authors. An example of this is the use of "linear functional" when "bounded linear functional" was what was meant. ...

I myself like the book very much. The book has many strong features, including the selection of topics, the notes, the exercises, the applications, the chapter on locally convex spaces, etc. Even the students who had no knowledge of (or interest in) physics appreciated the fact that the analysis they were studying is useful outside of mathematics, and this is an important motivating factor.

While the book is perhaps a little tough for first year students, it ought to be fine for second-year students who have a solid year of real analysis, including a little functional analysis (such as in W Rudin's Real and Complex Analysis). I'm glad I have the book and I expect to refer to it more than once.

2.6. Review by: Mitchell J Feigenbaum.
Transport Theory and Statistical Physics 2 (4) (1972), 373-375.

Mathematical physics texts have almost exclusively been treatises concerned with imparting to the student a working knowledge of the techniques of applied mathematics pertaining to equations of physics - ordinary and partial differential equations and integral equations. The techniques, of course, are power series and singularity structure in the setting of the theory of functions of a single complex variable. While this is adequate for just its purpose - i.e., recognising and solving the usual equations of continuum and quantum mechanics - it leaves most physicists with a marginal preparation for those techniques of more modern mathematics that one has begun to see in modern physics. At a more fundamental level, these texts hardly discourse on the Hilbert space setting of non-relativistic quantum mechanics - and invariably leave wholly untouched questions of unbounded operators. This latter topic, far from being pathological, is of course the very core of quantum mechanics. Not only is the theory of Lie groups never presented in any full way, but even the relation of Lie groups to solutions of differential equations is never seen. And any algebra beyond linear algebra and most elementary group theory is regarded as sophisticated beyond mention.

This is all well and good, were it true that the classical equations and the ability to do more imaginative contour integrals were the full substance of modern physics. But, with statistical mechanics being fundamentally measure theoretic and partly algebraic in nature; relativistic quantum mechanics deeply mired in operator valued distributions with anomalies abounding when too usual and naive a Dirac formalism is employed; general relativity "living" on an analytic manifold; and particle physics quite generally seeking out new paths and yearning for a broader, less trammelled setting; it is difficult to comprehend why "modern" mathematics has remained so far from the grasp of most physicists, when even a year's worth of study at the graduate level can at least give a physicist the freedom to decide

3.1. From the Publisher.

This volume will serve several purposes: to provide an introduction for graduate students not previously acquainted with the material, to serve as a reference for mathematical physicists already working in the field, and to provide an introduction to various advanced topics which are difficult to understand in the literature. Not all the techniques and application are treated in the same depth. In general, we give a very thorough discussion of the mathematical techniques and applications in quantum mechanics, but provide only an introduction to the problems arising in quantum field theory, classical mechanics, and partial differential equations. Finally, some of the material developed in this volume will not find applications until Volume III. For all these reasons, this volume contains a great variety of subject matter. To help the reader select which material is important for him, we have provided a "Reader's Guide" at the end of each chapter.

3.2. From the Preface.

This volume continues our series of texts devoted to functional analysis methods in mathematical physics. In Volume I we announced a table of contents for Volume II. However, in the preparation of the material it became clear that we would be unable to treat the subject matter in sufficient depth in one volume. Thus, the volume contains Chapters IX and We expect that a third volume will appear in the near future containing the rest of the material announced as "Analysis of Operators." We hope to continue this series with an additional volume on algebraic methods.

3.3. From the Introduction.

Most texts in functional analysis suffer from a serious flaw that is shared to an extent by Volume 1 of Methods of Modern Mathematical Physics. Namely, the subject is presented as an abstract, elegant corpus generally divorced from applications. Consequently, students who learn from these texts are ignorant of the fact that almost all deep ideas in functional analysis have their immediate roots in "applications," either to classical areas of analysis such as harmonic analysis or partial differential equations, or to another science, primarily physics. For example, it was classical electromagnetic potential theory that motivated Fredholm's work on integral equations and thereby the work of Hilbert, Schmidt, Weyl, and Riesz on the abstractions of Hilbert space and compact operator theory. And it was the impetus of quantum mechanics that led von Neumann to his development of unbounded operators and later to his work on operator algebras.

More deleterious than historical ignorance is the fact that students are too often misled into believing that the most profitable directions for research in functional analysis are the abstract ones. In our opinion, exactly the opposite is true. We do not mean to imply that abstraction has no role to play. Indeed, it has the critical role of taking an idea from a concrete situation and, by eliminating the extraneous notions, making the idea more easily understood as well as applicable to a broader range of situations. But it is the study of specific applications and the consequent generalisations that have been the more important, rather than the consideration of abstract questions about abstract objects for their own sake.

This volume contains a mixture of abstract results and applications, while the next contains mainly applications. The intention is to offer the readers of the whole series a properly balanced view

We hope that this volume will serve several purposes: to provide an introduction for graduate students not previously acquainted with the material, to serve as a reference for mathematical physicists already working in the field, and to provide an introduction to various advanced topics that are difficult to understand in the literature. Not all the techniques and applications are treated in the same depth. In general, we give a very thorough discussion of the mathematical techniques and applications in quantum mechanics, but provide only an introduction to the problems arising in quantum field theory, classical mechanics, and partial differential equations. Finally, some of the material developed in this volume will not find application until Volume 3. For all these reasons, this volume contains a great variety of subject matter. To help the reader select which material is important, we have provided a "Reader's Guide" at the end of each chapter.

As in Volume 1, each chapter contains a section of notes. The notes give references to the literature and sometimes extend the discussion in the text. Historical comments are always limited by the knowledge and prejudices of the authors, but in mathematics that arise directly from applications, the problem of assigning credit is especially difficult. Typically, the history is in two stages: first, a specific method (typically difficult, computational, and sometimes non-rigorous) is developed to handle a small class of problems. Later, it is recognised that the method contains ideas that can be used to treat other problems, so the study of the method itself becomes important. The ideas are then abstracted, studied at the abstract level, and the techniques systematised. With the newly developed machinery, the original problem becomes an easy special case. In such a situation, it is often not completely clear how many of the mathematical ideas were already contained in the original work. Further, how one assigns credit may depend on whether one first learned the technique in the old computational way or in the new, easier but more abstract way. In such situations, we hope that the reader will treat the notes as an introduction to the literature and not as a judgment of the historical value of the contributions in the papers cited.

3.4. Review by: Raul R Chernoff.
Mathematical Reviews MR0493420 (58 #12429b).

Volume II, "Fourier analysis, self-adjointness", contains two chapters. Chapter IX, "The Fourier transform", is a modern introduction to Fourier analysis. The central properties are developed efficiently via Schwartz spaces and distributions. There is a good discussion of interpolation methods, including the Riesz-Thorin and Marcinkiewicz theorems (the latter without proof, but with applications). Among the applications to partial differential equations is a proof of the Malgrange-Ehrenpreis theorem on fundamental solutions. A number of physical applications find their place in the course of a discussion of Wightman's axiomatic approach to quantum field theory. The long Chapter X, "Selfadjointness and the existence of dynamics", is largely devoted to developing conditions which guarantee that a symmetric operator is essentially selfadjoint. Prime examples are the Hamiltonian operators of nonrelativistic quantum mechanics, but there are also applications to quantum field theory. Among the topics discussed are the fundamental theorem of Kato-Rellich, quadratic form perturbations, a recent positivity method of Kato's, and analytic vector techniques. A short introduction to semigroup theory is followed by a section on hypercontractive semigroups and their perturbations - which has important applications to field theory. The chapter concludes with some miscellaneous topics: the Feynman-Kac formula and Wiener measure, time-dependent Hamiltonians, and the Hilbert space approach to classical mechanics. A useful table summarises the various methods and their applications - e.g., five different ways to cope with the anharmonic oscillator Hamiltonian.

3.5. Review by William Farris.
Bulletin of the American Mathematical Society 2 (3) (1980), 522-530.

See review 2.4. above.

4.1. From the Publisher.

Topics covered include: overview; classical particle scattering; principles of scattering in Hilbert space; quantum scattering; long range potentials; optical and acoustical scattering; the linear Boltzmann equation; nonlinear wave equations; spin wave scattering; quantum field scattering; and phase space analysis of scattering and spectral theory.

4.2. From the Introduction.

Scattering theory is the study of an interacting system on a scale of time and/or distance which is large compared to the scale of the interaction itself. As such, it is the most effective means, sometimes the only means, to study microscopic nature. To understand the importance of scattering theory, consider the variety of ways in which it arises. First, there are various phenomena in nature (like the blue of the sky) which are the result of scattering. In order to understand the phenomenon (and to identify it as the result of scattering) one must understand the underlying dynamics and its scattering theory. Second, one often wants to use the scattering of waves or particles whose dynamics on knows to determine the structure and position of small or inaccessible objects. For example, in x-ray crystallography (which led to the discovery of DNA), tomography, and the detection of underwater objects by sonar, the underlying dynamics is well understood. What one would like to construct are correspondences that link, via the dynamics, the position, shape, and internal structure of the object to the scattering data. Ideally, the correspondence should be an explicit formula which allows one to reconstruct, at least approximately, the object from the scattering data. The main test of any proposed particle dynamics is whether one can construct for the dynamics a scattering theory that predicts the observed experimental data. Scattering theory was not always so central the physics. Even thought the Coulomb cross section could have been computed by Newton, had he bothered to ask the right question, its calculation is generally attributed to Rutherford more than two hundred years later. Of course, Rutherford's calculation was in connection with the first experiment in nuclear physics.

Scattering theory is so important for atomic, condensed matter, and high energy physics that an enormous physics literature that has grown up. Unfortunately, the development of the associated mathematics has been much slower. This is partly because the mathematical problems are hard, but also because a lack of communication often makes it difficult for mathematicians to appreciate the many beautiful and challenging problems in scattering theory. The physics literature, on the other hand, is not entirely satisfactory because of the many heuristic formulas and ad hoc methods. Much of the physics literature deals with the "time-independent" approach to scattering theory because the time-independent approach provides powerful calculational tools. We feel that to use the time-independent formulas, one must understand them in terms of and derive them from the underlying dynamics. Therefore, in this book, we emphasise scattering theory as a time-dependent phenomenon, in particular, as a comparison between the interacting and free dynamics. This approach leads to a certain imbalance in our presentation since we therefore emphasise large times rather than large distances. However, as the reader will see, there is considerable geometry lurking in the background.

The scattering theories in branches of physics as different as classical mechanics, continuum mechanics, and quantum mechanics have in common the two foundational questions of the existence and completeness of the wave operators. These two questions are, therefore, our main object of study in individual systems and are the unifying theme that runs throughout the book. Because we treat so many different systems, we do not carry the analysis much beyond the construction and completeness of the wave operators, except in two-body quantum scattering, which we develop in some detail. However, even there, we have not been able to include such important topics as Regge theory, inverse scattering, and double dispersion relations

Since quantum mechanics is a linear theory, it is not surprising that the heart of the mathematical techniques is the spectral analysis of Hamiltonians. Bound states (corresponding to point spectra) of the interaction Hamiltonian do not scatter, while states of the absolutely continuous spectrum do. The mathematical property that distinguishes these two cases (and that connects the physical intuition with the mathematical formulation) is the decay of the Fourier transform of the corresponding spectral measures. The case of the singular continuous spectrum lies between the point spectra and the crucial (and often hardest) step in most proofs of asymptotic completeness. This is the proof that the interacting Hamiltonian has no singular continuous spectrum. Conversely, one of the best ways of showing that a self-adjoint operator has no singular continuous spectrum is to show that it is the interaction Hamiltonian of a quantum system with complete wave operators. This deep connection between scattering theory and spectral analysis shows the artificiality of the division of matter into Volumes III and IV. We have, therefore, preprinted at the end of this volume three sections on the absence of continuous singular spectrum from Volume IV.

While we were reading the galley proofs for this volume, V Enss introduced new and beautiful methods into the study of quantum-mechanical scattering. Enss's paper is of interest not only for what it proves but also for the future direction that it suggests. In particular, it seems likely that the methods will provide strong results in the theory of multiparticle scattering. We have added a section at the end of this chapter (Section XI.17) to describe Enss's method in the two-body case. We would like to thank Professor Enss for his generous attitude, which helped us to include this material

The general remarks about notes and problems made in earlier introductions are applicable here with one addition: the bulk of the material presented in this volume is from advanced research literature, so many of the problems are quite substantial. Some of the starred problems summarise the contents of research papers!

4.3. Review by: Raul R Chernoff.
Mathematical Reviews MR0529429 (80m:81085).

This third volume in the authors' series on functional analysis and its applications to contemporary mathematical physics deals primarily with the time-dependent approach to scattering theory. As the authors point out, the spectral properties of the Hamiltonian operator lie at the heart of this theory; in particular a fundamental problem in quantum scattering, related to "asymptotic completeness", is to establish the absence of a singular continuous spectrum. Thus Volume IV is an essential companion to the present volume, which for the reader's convenience includes reprints of several particularly germane sections of Volume IV.

4.4. Review by William Farris.
Bulletin of the American Mathematical Society 2 (3) (1980), 522-530.

See review 2.4. above.

4.5. Review by: Roger G Newton.
American Scientist 68 (2) (1980), 206.

Scattering theory has undergone extensive mathematical development during the last 25 years, and it is still growing. A physicist may be tempted to say that parts of this development are of no practical interest; they are, for him or her, merely ornamental. But then, many an engineer feels the same about some of the physicist's favourite theories. There is no denying that some results obtained by rigorous mathematics have led to important physical insights. Relevant examples from scattering theory are the optical theorem, Regge poles, resonance theory, solitons, and three-body reactions.

This book shows with great flair that certain aspects of scattering theory have become part of mathematics, and it treats them as such. In spite of various intuitive physical introductions and occasional discussions that show that the authors have an excellent understanding of the underlying physics, this is not a physics book with attention to mathematical rigor; it is an excellent mathematics text that treats scattering theory as a chapter in functional analysis, examining it in various physical contexts, such as classical particle scattering, quantum particle scattering, electromagnetic, acoustic, spin-wave, and quantum-field scattering.

More specifically, scattering theory is regarded as a special branch and application of the theory of the continuous spectrum of self-adjoint operators. It is the existence and completeness of the Moller wave operator that occupies most of the authors' attention, and their rigorous proofs often centre on, or are equivalent to, proofs of the absence of a singular continuous spectrum, a proposition which physicists usually take for granted. Perhaps the clearest indication that this volume is a book of mathematics is that on at least one occasion the authors present three (instructively) different proofs of the same theorem.

I strongly recommend this book to mathematicians or graduate students who would like to see an application of spectral theory, as well as to mathematical physicists working in rigorous scattering theory. Those who want to learn the physics or the calculational tools of collision theory will be better served by some of the other available treatises.

5.1. From the Preface.

With the publication of Volumes III and IV we have completed our presentation of the material which we originally planned as "Volume II" at the time of publication of Volume I. We originally promised the publisher that the entire series would be completed nine months after we submitted Volume I. Well! We have listed the contents of future volumes below. We are not foolhardy enough to make any predictions.

5.2. From the Introduction.

The first step in the mathematical elucidation of a physical theory must be the solution of the existence problem for the basic dynamical and kinematical equations of the theory. Once that is accomplished, one would like to find general qualitative features of these solutions and also to study in detail specific special systems of physical interest.
...
Nonrelativistic quantum mechanics is often viewed by physicists as an area whose qualitative structure, especially on the level treated here, is completely known. It is for this reason that a substantial fraction of the theoretical physics community would regard these volumes as exercises in pure mathematics. On the contrary, it seems to us that much of this material is an integral part of modern quantum theory. To take a specific example, consider the question of showing the absence of the singular continuous spectrum and the question of proving asymptotic completeness for the purely Coulombic model of atomic physics. The former problem was solved affirmatively by Balslev and Combes in 1970, the latter is still open. Many physicists would approach these questions with Goldberger's method: "The proof is by the method of reductio ad absurdum. Suppose asymptotic completeness is false. Why that's absurd! Q.E.D." Put more precisely: If asymptotic completeness is not valid, would we not have discovered this by observing some bizarre phenomena in atomic or molecular physics? Since physics is primarily an experimental science, this attitude should not be dismissed out of hand and, in fact, we agree that it is extremely unlikely that asymptotic completeness fails in atomic systems. But, in our opinion, theoretical physics should be a science and not an art and, furthermore, one does not fully understand a physical fact until one can derive it from first principles. Moreover, the solution of such mathematical problems can introduce new methods of calculational interest (for example, Faddeev's treatment of completeness in three-body systems and the application of his ideas in nuclear physics) and can provide important elements of clarity ...

5.3. Review by: Raul R Chernoff.
Mathematical Reviews MR0493421 (58 #12429c).

Volume IV, "Analysis of operators", takes up the detailed study of the spectra and eigenfunctions of Hamiltonian operators. ... Chapter XII, "Perturbation of point spectra", introduces regular perturbation theory. Here the main result is the Kato-Rellich theorem, which gives useful conditions guaranteeing the convergence of the formal perturbation series found in the physics books. The authors go on to asymptotic and summability methods that can be effective when the perturbation series diverges. This short chapter concludes with a rigorous discussion of the sometimes rather misty notion of "resonances". The bulk of Volume IV is taken up by Chapter XIII, "Spectral analysis". This chapter surveys a number of important methods for obtaining qualitative and quantitative information about spectra and eigenfunctions. Sample quantitative question: Is the discrete spectrum finite, and if so how can one estimate the number of bound states? Some qualitative questions: Is there a ground state, and is it nondegenerate? Is there any singular continuous spectrum? This is one of the chief problems in scattering theory; the answer is supposed to be no for physically reasonable potentials. Much of the chapter is devoted to the analysis of this problem.

5.4. Review by William Farris.
Bulletin of the American Mathematical Society 2 (3) (1980), 522-530.

See review 2.4 above.

6.1. From the publisher.

Barry Simon's book both summarises and introduces the remarkable progress in constructive quantum field theory that can be attributed directly to the exploitation of Euclidean methods. During the past two years deep relations on both the physical level and on the level of the mathematical structure have been either uncovered or made rigorous. Connections between quantum fields and the statistical mechanics of ferromagnets have been established, for example, that now allow one to prove numerous inequalities in quantum field theory.

In the first part of the book, the author presents the Euclidean methods on an axiomatic level and on the constructive level where the traditional results of the $P(\phi)_{2}$ theory are translated into the new language. In the second part Professor Simon gives one of the approaches for constructing models of non-trivial, two-dimensional Wightman fields - specifically, the method of correlation inequalities. He discusses other approaches briefly.

Drawn primarily from the author's lectures at the Eidenössiehe Technische Hochschule, Zurich, in 1973, the volume will appeal to physicists and mathematicians alike; it is especially suitable for those with limited familiarity with the literature of this very active field.

6.2. From the Preface.

These lecture notes are mainly based on a series of lectures given at the Seminar for Theoretische Physik of the ETH/EPF - Zurich in the Spring of 1973. It is a great pleasure to thank the many people who helped in my efforts:

Klaus Hepp for inviting me to lecture at the ETH, Jean Lascoux (Ecole Polytechnique-Paris), Paul Urban (Schladming), Daniel Kastler (CNRS-Marseille), André Lichnierowicz (College de France), R Gerard (Strassbourg), and John Lewis (Institute for Advanced Studies-Dublin) for the opportunity to present a lecture series on this material; these "dress rehearsals" allowed me to experiment in many ways with the presentation of the material, David Ruelle and Walter Thirring for convincing me that the time was right for such written lecture notes, James Glimm, Arthur Jaffe, Ed Nelson, and Arthur Wightman for all they have taught me, Francesco Guerra and Lon Rosen for the joy of collaboration and for permission to use material we are still in the process of writing up, Miss R Hintermann for typing the bulk of the first draft (7.5 chapters). This was done during a ten-week period which was exceptionally gruelling for both of us! Mrs G Anderson and Mrs C Jones for the rest of the typing of the first draft and Mrs H Morris for the final typed copy, Arthur Wightman for his enthusiasm at publication of the notes (and for his enthusiasm in general!) ...

6.3. From the Introduction.

These lecture notes are intended to introduce the reader to Euclidean ideas in quantum field theory and then to develop one approach, the "correlation inequality" method, to the simplest model of an interacting quantum field theory, the $P(\phi)_{2}$ model of a self-coupled Bose field in two dimensional space-time. We have tried hard to make them accessible to non-trivial subsets of both the mathematics community and the physics community. We have emphasised the probabilistic Euclidean strategy toward $P(\phi)_{2}$ over the Hamiltonian strategy, which in the hands of Glimm and Jaffe dominated the period from 1964 to 1971 and which has played such an important role in shaping the Euclidean strategy. ...
...
We emphasise to the reader that these are lecture notes and are not intended as a polished product. In a field in the state of flux in which constructive quantum field theory was in 1973, the latter was almost impossible. Results that were discovered or recognised while I was in the process of lecturing tended to find their way into the next relevant place when their most natural point had already passed.

The difficulty of the material and the amount left to the reader is quite variable, but I have tried whenever possible to heed the country preacher's advice on sermons, "First, you tell 'em what you're going to tell 'em, then you tell 'em, then you tell 'em what you've told 'em."

6.4. Review by: Raymond Streater.
Mathematical Reviews MR0489552 (58 #8968).

This book is a phenomenon, and the author's power over detail is remarkable. The reader is able (with a bit of work, it is true) to enter the very difficult field of constructive quantum field theory up to the very latest and best results up to 1974. The book is suitable as a text for a postgraduate seminar for probabilists and analysts intent on some hard work.

Equally impressive is the topicality of the book (in 1974). The only thing missing at the time of publication was the cluster expansion of Glimm, Jaffe and Spencer. Consequent to its rapid publication and somewhat hasty writing, there are some misprints - the worst is that part of the statement of one theorem is missing.

6.5. Review by: I E Segal.
American Scientist 63 (6) (1975), 718.

Here is an enthusiastic, almost dithyrambic account of the most complete chapter to date in Constructive Quantum Field Theory. It is five years since the theory was put on an entirely sound basis and its cogency established in the case of two-dimensional scalar fields with a "spatial cutoff." Some fundamental questions of existence and unicity remain for this case, but there has nevertheless been a field day in the analysis of a number of the resulting clearly posed mathematical problems, by the methods of functional analysis, abstract probability theory, and statistical mechanics.

The theory originates in the reconciliation of the wave and corpuscular theories of light via the quantisation of the Maxwell equations. This is one of the most striking achievements of mathematical physics, but its nonlinear extension (to "quantum electrodynamics" and similar areas) remains, after 50 years, one of the most challenging scientific problems of the century. Is the nonlinear extension a specious illusion? or can it be mathematically well founded or otherwise used to make reliable, verifiable predictions? The mathematical progress of the past 25 years has opened up vast new areas of analysis and attained an encouraging momentum, without as yet having given a definitive answer to these questions.

Simon's book is an articulate and cultured, if somewhat partisan and turgid, treatment of one of the notable markers on (or at least near) the road to this end. The beautiful and powerful "euclidean" theory, due basically to Nelson (roughly, the felicitous application of the idea of replacing $t$ by $x√-1 t$, an idea that in a totally heuristic way goes back in quantum field theory to Schwinger), has been a major component in the intensive development in the past few years in scalar two-dimensional quantum field theory. The present exposition of it is scientific journalism of very high order and one of the most readable introductions to the whole subject, even if (or possibly partly because) it emphasizes developments which may be intrinsically limited to the models treated.

7.1. From the Publisher.

These expository lectures contain an advanced technical account of a branch of mathematical analysis. In his own lucid and readable style the author begins with a comprehensive review of the methods of bounded operators in a Hilbert space. He then goes on to discuss a wide variety of applications including Fredholm theory and more specifically his own specialty of mathematical quantum theory. included also are an extensive and up-to-date list of references enabling the reader to delve more deeply into this topical subject.

7.2. From the Preface.

Several years ago, I was working simultaneously on three problems: one involved scattering a quantum mechanical particle from a very singular repulsive core, one involved bounds on the number of negative eigenvalues ​​of $-\Delta + \lambda V$ with the correct behaviour as $\lambda \rightarrow ∞$, and the third involved the structure of the two-dimensional Yukawa quantum field theory. The physics and the fundamental mathematical structure of these problems are quite different. But it turned out that the technical tools needed to solve the problems were remarkably similar, so much so that at times I couldn't keep straight which one I was thinking about. Since that time, I have had a great respect and use for a subject that might be called "the hard analysis of compact operators in Hilbert space." I discovered that many of the ideas that I grew so fond of had already been developed by Russian mathematicians and mathematical physicists, particularly the group around M S Birman.

In these lectures, I wish to describe the main ideas and illustrate the tools in a set of specific problems. I am a firm believer in the principle that ideas in analysis should be valued largely by their applicability to other parts of mathematics, so I have included many applications chosen from my own specialty of mathematical physics, especially quantum theory. However, I have sufficient faith in these tools that I don't doubt that I would have many applications if I worked in some other area of ​​analysis I warn the reader that there is some overlap with pedagogical presentations I have given elsewhere of bits and pieces of this material and that virtually nothing I have to say here is not already in the research literature. For beautiful presentations of some of the material from a somewhat different viewpoint I highly recommend the monograph of Gohberg-Krein and Ringrose. Many of the results of the Birman school are summarised in the lecture notes of Birman and Solomjak which have recently been translated.

7.3. Review by: Michael Demuth.
Mathematical Reviews MR0541149 (80k:47048).

The theory of trace ideals is reviewed. The main ideas are described and illustrated by applications chosen from mathematical physics. No new results are given. The postgraduate level is useful for introducing a wider audience to the trace ideal theory. For applying this theory a lot of estimations and trace class criteria are given.

8.1. From the Publisher.

The main theme of this book is the "path integral technique" and its applications to constructive methods of quantum physics. The central topic is probabilistic foundations of the Feynman-Kac formula. Starting with main examples of Gaussian processes (the Brownian motion, the oscillatory process, and the Brownian bridge), the author presents four different proofs of the Feynman-Kac formula. Also included is a simple exposition of stochastic Ito calculus and its applications, in particular to the Hamiltonian of a particle in a magnetic field (the Feynman-Kac-Ito formula). Among other topics discussed are the probabilistic approach to the bound of the number of ground states of correlation inequalities (the Birman-Schwinger principle, Lieb's formula, etc.), the calculation of asymptotics for functional integrals of Laplace type (the theory of Donsker-Varadhan) and applications, scattering theory, the theory of crushed ice, and the Wiener sausage. Written with great care and containing many highly illuminating examples, this classic book is highly recommended to anyone interested in applications of functional integration to quantum physics. It can also serve as a textbook for a course in functional integration.

8.2. From the Preface.

In the summer of 1977 I was invited to lecture in the Troisième Cycle de la Suisse Romande, a consortium of four universities in the French-speaking part of Switzerland. There was some discussion of the topic about which I might speak. Since I seem fated to be the apostle of probability to Swiss physics (see [258]), we agreed on the general topic of "path integral techniques." I decided to limit myself to the well-defined Wiener integral rather than the often ill-defined Feynman integral. In preparing my lectures I was struck by the mathematical beauty of the material, especially some of the ideas about which I had previously been unfamiliar. I was also struck by the dearth of "expository" literature on the connection between Wiener integral techniques and their application to rather detailed questions in differential equations, especially those of quantum physics; it seemed that path integrals were an extremely powerful tool used as a kind of secret weapon by a small group of mathematical physicists. My purpose here is to rectify this situation. I hope not only to have made available new tools to practice mathematical physicists but also to have opened up new areas of research to probabilists.

8.3. From the Introduction.

It is fairly well known that one of Hilbert's famous list of problems is that of developing an axiomatic theory of mathematical probability theory (this problem could be said to have been solved by Khintchine, Kolmogorov, and Lévy), and also among the list is the "axiomatisation of physics." What is not so well known is that these are two parts of one and the same problem, namely, the sixth, and that the axiomatics of probability are discussed in the context of the foundations of statistical mechanics. Although Hilbert could not have known it when he formulated his problems, probability theory is also central to the foundations of quantum theory. In this book, I wish to describe a very different interface between probability and mathematical physics, namely, the use of certain notions of integration in function spaces as technical tools in quantum physics. Although Nelson has proposed some connection between these notions and foundational questions, we shall deal solely with their use to answer a variety of questions in conventional quantum theory ...

8.4. Review by: Vadim Aleksandrovich Malyshev.
Mathematical Reviews MR0544188 (84m:81066).

This excellent book is highly recommended to anyone concerned with functional integration or the constructive approach in physics. It can also serve as an introductory course.

The first main theme of the book is the probabilistic foundation of the Feynman-Kac formula. After the concise exposition of the main tools of abstract Gaussian processes the author constructs three processes: the Brownian motion, the Ornstein-Uhlenbeck process (oscillatory process) and the Brownian bridge. ...

The second main theme is a probabilistic proof of the bounds for the number of bound states: the Birman-Schwinger principle, Lieb's formula, phase space bounds and others. There is also a proof of the stability of matter. The third theme is the applications of correlation inequalities, which first appeared in statistical mechanics, $P(\phi)_{1}$ and related processes. The fourth theme concerns the calculation of logarithmic asymptotics for functional integrals of Laplace type - the theory of Donsker-Varadhan in connection with the Gibbs variational principle from statistical mechanics. Unfortunately deep related results of A Ventcel' and M Freidlin are not included, even in the bibliography. Finally, the author gives applications to scattering theory, the theory of crushed ice and the Wiener sausage.

Thus the book mainly deals with application of powerful probabilistic techniques, which were developed recently in statistical mechanics and quantum field theory, to one-dimensional problems connected with Wiener processes. The final short introduction to some multidimensional random field problems such as the sine-Gordon transformation emphasises this relationship with statistical mechanics and quantum field theory.

8.5. Review by: James Glimm.
American Scientist 68 (2) (1980), 228.

Functional integrals arise in a variety of problems in mathematics, physics, and chemistry. The central theme of these applications and of this book is the Feynman-Kac formula.

Modern functional analysis and operator theory dominate Simon's presentation, in contrast to most other accounts of function space integrals, which emphasise the probabilistic aspects of the subject. Simon's book offers an interesting and healthy range of applications, drawn from his interests in quantum mechanics and including the classical limit, ground state perturbation theory and the stability of-matter problem. The book is written at the level of a graduate mathematics text.

9.1. From the Preface.

This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modern mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained, although there are occasional references to later volumes. We have included a few applications when we thought that they would provide motivation for the reader. Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, modern physics, and partial differential equations.

This revised and enlarged edition differs from the first in two major ways. First, many colleagues have suggested to us that it would be helpful to include some material on the Fourier transform in Volume I so that this important topic can be conveniently included in a standard functional analysis course using this book. Thus, we have included in this edition Sections IX.1, IX.2, and part of IX.3 from Volume II and some additional material, together with relevant notes and problems. Secondly, we have included a variety of supplementary material at the end of the book. Some of these supplementary sections provide proofs of theorems in Chapters II-IV which were omitted in the first edition. While these proofs make Chapters II-IV more self-contained, we still recommend that students with no previous experience with this material consult more elementary texts. Other supplementary sections provide expository material to aid the instructor and the student (for example, "Applications of Compact Operators"). Still other sections introduce and develop new material (for example, "Minimisation of Functionals").

9.2. From the Introduction.

Mathematics has its roots in numerology, geometry, and physics. Since the time of Newton, the search for mathematical models for physical phenomena has been a source of mathematical problems. In fact, whole branches of mathematics have grown out of attempts to analyse particular physical situations. An example is the development of harmonic analysis from Fourier's work on the heat equation.

Although mathematics and physics have grown apart in this century, physics has continued to stimulate mathematical research. Partially because of this, the influence of physics on mathematics is well understood. However, the contributions of mathematics to physics are not as well understood. It is a common fallacy to assume that mathematics is important for physics only because it is a useful tool for making computations. Actually, mathematics plays a more subtle role, which in the long run is more important. When a successful mathematical model is created for a physical phenomenon - that is, a model that can be used for accurate computations and predictions - the mathematical structure of the model itself provides a new way of thinking about the phenomenon Put slightly differently, when a model is successful, it is natural to think of physical quantities in terms of the mathematical objects that represent them and to interpret similar or secondary phenomena in terms of the same model. Because of this, an investigation of the internal mathematical structure of the model can alter and enlarge our understanding of the physical phenomenon. Of course, the outstanding example of this is Newtonian mechanics, which provided such a clear and coherent picture of celestial motions that it was used to interpret practically all physical phenomena. The model itself became central to an understanding of the physical world, and it was difficult to give it up in the late nineteenth century, even in the face of contradictory evidence. A more modern example of this influence of mathematics on physics is the use of group theory to classify elementary particles.

9.3. Review by: Editors.
Mathematical Reviews MR0751959 (85e:46002).

The first edition has been reviewed (see above). The present edition is enlarged in two major ways: (1) There is a chapter on the Fourier transform (namely, Sections IX.1, IX.2 and parts of IX.3 from Volume II, and some additional material). (2) There is a variety of supplementary material at the end of the book. Some of these sections provide proof of theorems in Chapters II-IV which were omitted in the first edition, while others introduce and develop new material.

10.1. From the Publisher.

A complete understanding of Schrödinger operators is a necessary prerequisite for unveiling the physics of nonrelativistic quantum mechanics. Furthermore recent research shows that it also helps to deepen our insight into global differential geometry. This monograph written for both graduate students and researchers summarises and synthesises the theory of Schrödinger operators emphasising the progress made in the last decade by Lieb, Enss, Witten and others. Besides general properties, the book covers, in particular, multiparticle quantum mechanics including bound states of Coulomb systems and scattering theory, quantum mechanics in constant electric and magnetic fields, Schrödinger operators with random and almost periodic potentials and, finally, Schrödinger operator methods in differential geometry to prove the Morse inequalities and the index theorem.

10.2. From the Preface by Barry Simon.

In the summer of 1982, I gave a course of lectures in a castle in the small town of Thurnau outside Bayreuth, West Germany, whose university hosted the lecture series. The Summer School was supported by the Volkswagen Foundation and organized by Professor C Simader, assisted by Dr H Leinfelder. I am grateful to these institutions and individuals for making the school, and thus this monograph, possible.

About 40 students took part in a gruelling schedule involving about 45 hours of lectures spread over eight days! My goal was to survey the theory of Schrödinger operators emphasising recent results. While I would emphasise that one was not supposed to know all of Volumes 1-4 of Reed and Simon (as some of the students feared!), a strong grounding in basic functional analysis and some previous exposure to Schrödinger operators was useful to the students and will be useful to the reader of this monograph

Loosely speaking, Chaps. 1-11 of this monograph represent the "notes" of those lectures taken by three of the "students" who were there. While the general organisation does follow mine, I would emphasise that what follows is far from a transcription of my lectures. Even with 45 hours, many details had to be skipped, and quite often Cycon, Froese, and Kirsch had to flesh out some rather dry bones. Moreover, they have occasionally rearranged my arguments, replaced them with better ones, and even corrected some mistakes!

Some results, such as Lieb's theorem (Theorem 3.17), that were relevant to the material of the lectures but appeared during the preparation of the monograph have been included

Chapter 11 of the lectures concerns some beautiful ideas of Witten reducing the Morse inequalities to the calculation of the asymptotics of eigenvalues ​​of cleverly chosen Schrödinger operators (on manifolds) in the semiclassical limit. When I understood the supersymmetric proof of the Gauss-Bonnet-Chern theorem (essentially due to Patodi) in the summer of 1984, and, in particular, using Schrödinger operator ideas found a transparent approach to its analytic part, it seemed natural to combine it with Chapter 11, and so I wrote a twelfth chapter. Since I was aware that Chapters 11 and 12 would likely be of interest to a wider class of readers with less of an analytic background, I have included in Chapter 12 some elementary material (mainly according to Sobolev estimates) that have been freely used in earlier chapters

10.3. Review by: Review by: Michael Demuth.
Mathematical Reviews MR0883643 (88g:35003).

This book gives an extensive overview of recent developments in the theory of Schrödinger operators, emphasising the progress of recent years. At the beginning of each chapter its aim is explained and the most important literature for the topics considered is given. Moreover, in many cases, the objectives of the theorems are described so that the content is more intelligible to the reader. For every proof, references to the literature are cited. Many proofs are only sketched or indicated. So the book is not self-contained in any detail. But it is very useful for all scientists who wish to obtain information about some recent results in the theory of Schrödinger operators.

Starting with the selfadjointness problem for Schrödinger operators with scalar and vector potentials, Stummel and Kato class potentials are considered. Properties of the eigenfunctions are studied by investigating semigroup $L^{p}$-properties. The geometry of the phase space is used to consider bound states and the essential spectrum in $N$-body systems. For instance, Lieb's result on the number of electrons bound by the nuclei in many-atom systems is derived. Local commutator estimates, in particular, the Mourre estimate, are explained. They are used to control imbedded eigenvalues, to prove the absence of singularly continuous spectra, to show the nonexistence of positive eigenvalues in $N$-body systems. Quantum mechanical scattering is presented by means of the geometric methods due to Enss. The asymptotic completeness is proved for two-body systems. Some features of three-body systems are discussed. The scattering considerations are restricted to short-range potentials and time-dependent methods. Then a few aspects of Schrödinger operators with magnetic fields (gauge invariance, spectral properties, supersymmetric Hamiltonians, index theorem) and with electric fields (e.g. time evolution and spectral properties for systems with constant and time-dependent Stark fields) are considered. An overview is given for the theory of dilation analyticity. The authors mention $N$-body-problems, resonances, computational aspects of complex scaling, and connections to Mourre's estimate. Basic problems, techniques and results in the theory of Jacobi matrices with random and almost periodic potentials are introduced to some extent (density of states, Lyapunov exponent, spectrum for the Anderson model, fascinating spectral properties for almost periodic Jacobi matrices). Finally, some applications of the Schrödinger operator theory to analysis on manifolds are discussed. Links between the semiclassical eigenvalue limit behaviour and the Morse inequalities are described. Moreover, Schrödinger operator methods are used to prove the index theorem.

11.1. From the Publisher.

A state-of-the-art survey of both classical and quantum lattice gas models, this two-volume work will cover the rigorous mathematical studies of such models as the Ising and Heisenberg, an area in which scientists have made enormous strides during the past twenty-five years. This first volume addresses, among many topics, the mathematical background on convexity and Choquet theory, and presents an exhaustive study of the pressure including the Onsager solution of the two-dimensional Ising model, a study of the general theory of states in classical and quantum spin systems, and a study of high and low temperature expansions. The second volume will deal with the Peierls construction, infrared bounds, Lee-Yang theorems, and correlation inequality.

This comprehensive work will be a useful reference not only to scientists working in mathematical statistical mechanics but also to those in related disciplines such as probability theory, chemical physics, and quantum field theory. It can also serve as a textbook for advanced graduate students.

11.2. From the Introduction.

In 1979-80, my last year at Princeton (although I didn't know it at the time!), I gave a course on rigorous results in the statistical mechanics of discrete lattice models. In preparing the course, I realised that a twenty-year period of explosive growth in our knowledge seemed to be drawing to a close as the most approachable problems were solved. So it seemed like a good time to think about a book summarising the state of our knowledge.

It was clear such a book would need to be comprehensive, even encyclopaedic, since it was describing a mature subject. Little did I realise that within a short time the project would bifurcate into a two-volume plan, nor that it would stretch out for over ten years while I juggled other responsibilities and interests.

I was correct that the subject had matured. A book begun in 1969 would have had to be extensively rewritten as the seventies progressed but the changes/additions to this book as the process of writing got drawn out were not large. Indeed, the only result of the eighties comparable in depth to the major advances of the sixties and seventies were those of Fröhlich-Spencer on multiscale cluster expansions (which are to be discussed in volume 2).

I made several decisions about the book early on: I would restrict myself to "standard" lattice models which meant no spin glasses or field theories or... but I would include something about quantum systems. I'd freely use correlation inequalities in passing, even though their comprehensive treatment would wait for the end of volume 2.

Many of the "sexiest" topics - infrared bounds, the Fröhlich-Spencer theory, the Lee-Yang theorem, and correlation inequalities - have been postponed until volume 2. I hope that it won't take another fourteen years for it to appear but it is certainly going to take some time.

This has been a fun subject to write about as it has so many high points of great beauty while it requires little in the way of fancy functional analytic argument.

The earlier parts of the book were kicking around in manuscript form for many years and benefited from comments from many friends and colleagues to whom I am grateful.

11.3. Review by: Huzihiro Araki.
Mathematical Reviews MR1239893 (95a:82001).

This book contains a more or less comprehensive treatment of mathematical results in the statistical mechanics of lattice systems, both classical and quantum. Volume 1 covers, more or less, the general theory, postponing more specialised topics with detailed estimates to Volume 2, which is not expected to appear soon, according to the author's preface.

Chapter 1 deals with preliminaries, including commutative and noncommutative convexity inequalities and the Legendre transform. Chapter 2 deals with various aspects of the thermodynamic limit of the pressure function. Chapter 3 contains the general theory of equilibrium states and its thermodynamics for classical interaction and Chapter 4 contains the same subject for quantum interaction. Chapter 5 deals with high temperature and low density behaviour of equilibrium states, including uniqueness theorems, expansions and decay of correlations.

12.1. From the Publisher.

Barry Simon, I.B.M. Professor of Mathematics and Theoretical Physics at the California Institute of Technology, is the author of several books, including such classics as Methods of Mathematical Physics (with M Reed) and Functional Integration and Quantum Physics. This new book, based on courses given at Princeton, Caltech, ETH-Zurich, and other universities, is an introductory textbook on representation theory. According to the author, "Two facets distinguish my approach. First, this book is relatively elementary, and second, while the bulk of the books on the subject is written from the point of view of an algebraist or a geometer, this book is written with an analytical flavour".

The exposition in the book centres around the study of representation of certain concrete classes of groups, including permutation groups and compact semisimple Lie groups. It culminates in the complete proof of the Weyl character formula for representations of compact Lie groups and the Frobenius formula for characters of permutation groups. Extremely well tailored both for a one-year course in representation theory and for independent study, this book is an excellent introduction to the subject which, according to the author, is unique in having "so much innate beauty so close to the surface".

12.2. From the Introduction.

Oy! YABOGR - yet another book on group representations. The theory is so beautiful and so central to mathematics that it tends to draw authors like honey draws flies. Given the existence of several excellent monographs (among which I'd include Adams, Fulton-Harris, Samelson, and Serre), why do I feel this book is a worthy addition to the textbook literature on the subject?

I think two facets distinguish my approach. First, this book is relatively elementary; and second, while the bulk of books on the subject (and, in particular, the four quoted above) are written from the point of view of an algebraist or a geometer, this book is written with an analytical flavour

As for the elementary nature of the material, much of it is self-contained. A prior exposure to the notions of quotient groups and the isomorphism theorems is assumed; but, for example, I develop the necessary theory of algebraic integers in proving that the dimension of an irreducible representation divides the vector of the group. The material on finite groups (Chapters 1-VI) should be suitable for an upper-level undergraduate course, either as a separate course or as a supplement to an advanced algebra course.

The material on compact groups is a little more sophisticated, but I have discussed the needed calculus on manifolds and have even included an appendix on the basic theory of self-adjoint Hilbert-Schmidt operators, which is needed to prove the Peter-Weyl theorem.

However, there are a few places we need some facts from algebraic topology that we used without proof: for example, the exact sequence of a fibration

As for the analyst's point of view, much of the most profound work on group representations has been done by analysts Weyl, Gelfand, and Mackey come to mind. Indeed, this monograph bears a strong influence of George Mackey, from whom I first learned much of the material thirty years ago.

The analyst's approach can be seen in several places: for example, the last three chapters discuss the structure and representations of compact groups, not the representations of the semisimple Lie algebras. The two are closely related, but the former is more elementary and decidedly less algebraic. In this regard, the discussion is closest to that of Adams.

A critical role is played by the fact that for compact groups, it is easy to show that any (finite-dimensional, continuous) representation supports an invariant positive definite inner product. This immediately implies that on the Lie algebra, the adjoint representation is by matrices which are skew-adjoint in a suitable inner product, so the operators are semisimple and the Killing form is (strictly) negative definite. This replaces pages of algebraic minutiae.

A good example of this philosophy is the proof in Chapter VIII that all maximal tori in a compact Lie group are conjugate and the union of all the maximal tori is the entire group. The standard proofs either go through the conjugacy of the Cartan subalgebras and a considerable additional argument, or else use Weil's approach of using the Lefschitz fixed point theorem, a sledgehammer for what is a rather simple result. Instead, I use a simple argument inductive in the dimension of G. The first 90% of the proof is that used by Varadarajan, but at a critical point, he appeals to the structure theory of the semisimple Lie algebras, which requires tens of pages of careful algebraic argument. Instead, I use the existence of an invariant inner product for the adjoint representation.

This is one of dozens of places where the proofs are the ones I found while polishing the book. Nevertheless, I am not so naive as to think that there are any proofs here that don't appear somewhere in the literature, which is vast. But I do claim a coherent, elementary approach.

Individual chapters begin with brief summaries of what they contain. Chapter I sets the stage, focusing on counting principles as a leitmotif. The high point is the Klein-Weyl determination of the finite subgroups of three-dimensional rotations.

Chapters II-VI discuss the representations of finite groups. Chapters II-III develop the general theory, and Chapters IV-VI, the representations of specific families of groups: Abelian and Clifford groups in Chapter IV, semidirect products in Chapter V, and permutation groups in Chapter VI.

The final three chapters discuss the representations of compact groups, primarily compact lie groups. Chapter VII discusses the general theory of lie groups and the analogies of the results of Chapter III. Chapter VIII discusses the structure theory of compact lie groups: maximal tori, roots, and the Weyl group. It is preparation for the final chapter, which presents Weyl's theory of the representations of classical groups. The final section draws together the two halves of the book in a fascinating way by providing a proof of the Frobenius character formula for the permutation group. By focusing on finite and compact groups, we can present the basics completely. Any attempt to go beyond this would yield a multiple-volume work (as it has in other cases!). Nevertheless, when I've given this as a one-year graduate course, I have spent five weeks discussing related topics from the representation theory of noncompact groups.

This book is based on a course I first gave at Princeton in the mid-1970s. Over the ensuing twenty years, I have given the course roughly a half-dozen times at Princeton or Caltech, and each time additional polish was added. I am grateful to all the students in those courses for the feedback and insight they provided.

Parts of the actual manuscript were written during stays at the ETH-Zurich, Hebrew University, and the Technion. I appreciate their hospitality ...
...
I hope you will enjoy this book. I can't think of any other course of mathematics with so much innate beauty so close to the surface.

12.3. Review by: Antoni Wawrzynczyk.
Mathematical Reviews MR1363490 (97c:22001).

Although not divided explicitly, the book consists of two parts which should be considered separately. The first one is concerned with the theory of representations of finite groups; it contains 6 chapters and 120 pages. The second part, devoted to the theory of compact Lie groups, contains 3 chapters and 135 pages. Taking into account that the subject of this latter part is much more extensive and complicated, it is obvious that the author has had to apply a different approach in attempting to cover it in almost the same space.

In a concise form one can say that, while the first part can be considered as a complete and self-contained introduction to finite group representations, the second one presents selected topics of the theory of compact groups and their representations.
...
The author's promise to give more analytic flavour to the theory is kept only in the part concerning the structure of the compact Lie groups. Surprisingly, the algebraic parts of the book seem to be more complete and better organised.

The theory of representations of groups is nowadays a very extensive area. Textbooks presenting particular topics of the theory are very desirable. In particular this book can be recommended as a base for courses about representations of finite groups and finite-dimensional representations of Lie groups.

13.1. From the Publisher.

This two-part volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators.

Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by z (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line.

The book is suitable for graduate students and researchers interested in analysis.

13.2. Contents.

Chapter 1. The Basics
Chapter 2. Szegő's Theorem
Chapter 3. Tools for Geronimus' Theorem
Chapter 4. Matrix Representations
Chapter 5. Baxter's Theorem
Chapter 6. The Strong Szegő Theorem
Chapter 7. Verblunsky Coefficients With Rapid Decay
Chapter 8. The Density of Zeros

13.3. From the Introduction.

My first real exposure to orthogonal polynomials on the unit circle came in connection with my role as an editor of Communications in Mathematical Physics, in particular, with a submission to Golinskii and Nevai. My section in Communications in Mathematical Physics had accepted an earlier paper on the subject by Geronimo and Johnson, but at the time I had not paid close attention to it. I personally knew both those authors and the paper seemed to be studying some kind of general difference equation, so I sent it to an appropriate referee and followed the recommendation. The Golinskii-Nevai paper was different. I knew of Nevai's reputation but didn't know either author. The paper was clearly about orthogonal polynomials and I was reluctant, given Communications in Mathematical Physics' perpetual page crunch, to open up the journal to a subject that I worried might be disconnected to either physics or the areas that we normally cover. So I spent some time carefully reading the introduction, thinking about the issues, and skimming the rest. I realised this paper was one on spectral theory related to many papers in Communications in Mathematical Physics, so I was comfortable sending it out to a reference and following a positive recommendation.

13.4. Review by: Peter Larkin Duren.
Mathematical Reviews MR2105088 (2006a:42002a).

Orthogonal polynomials over an interval of the real line constitute a time-honoured subject that dates back to the 19th century in works of Legendre, Jacobi, Hermite, Laguerre, Chebyshev, and others. Developments were motivated largely by classical problems of mathematical physics. However, the first systematic studies of orthogonal polynomials over curves came only in the 1920s with work of Szegő. In 1921 he gave an account of polynomials orthogonal over a rectifiable Jordan curve and established a close connection with the Riemann mapping function of the exterior region. Around the same time, Szegő studied polynomials orthogonal over the unit circle with respect to weighted Lebesgue measure, mainly in connection with Toeplitz forms. The basic results were outlined in his classic book Orthogonal polynomials, first published in 1939. A subsequent book by Geronimus (1958) described further developments, but the theory for the circle appeared relatively narrow in scope.

Barry Simon has now opened new vistas with a monumental treatise that integrates old and new aspects of the subject and describes important connections with mathematical physics, especially with spectral theory of Schrödinger operators.
...
The book is long and is packed with information on a range of topics, but the author has made every effort to guide the reader toward a clear understanding of the material and its various interconnections. The result is an instant classic that will become a standard source for novices and veterans alike.

13.5. Review by: Jahresbericht der DMV.

Simon's work is not just a book about orthogonal polynomials but also about probability measures on one-dimensional Schrödinger operators and operator theory. It is extremely complex, multilayered, fascinating, and inspiring, while remaining very readable (even for advanced students). Without a doubt this monograph will become the standard reference for the theory of orthogonal polynomials on the unit circle for a long time to come.

13.6. Review by: Stefan Cobzas.
Studia Universitatis Babes-Bolyai, Mathematica.

Undoubtedly that ... this book will become a standard reference in the field tracing the way for future investigations on orthogonal polynomials and their applications. Combining methods from various areas of analysis (calculus, real analysis, functional analysis, complex analysis) as well as by the importance of the orthogonal polynomials in applications, the book will have a large audience including researchers in mathematics, physics, (and) engineering.

14.1. From the Publisher.

See 13.1. above.

14.2. Contents.

Chapter 9. Rakhmanov's theorem and related issues
Chapter 10. Techniques of spectral analysis
Chapter 11. Periodic Verblunsky coefficients
Chapter 12. Spectral analysis of specific classes of Verblunsky coefficients
Chapter 13. The connection to Jacobi matrices
Appendix A. Reader's guide: Topics and formulae
Appendix B. Perspectives
Appendix C. Twelve great papers
Appendix D. Conjectures and open question

14.3. From the Preface.

For an overview of the subject matter of these volumes, see Section 1.1 (in Part 1), and for a discussion of how this book came to be and my many debts to others, see the Preface to Part 1.

In many ways, these volumes are one large book broken in two, so I've used successive page and chapter numbering, starting in this volume where the other volume left off. The table of contents, list of notation, author index, and subject index cover both volumes. In this volume, there is a complete bibliography listing the references in the entire work.

I warn the reader of a personal quirk. I'm told that proper usage requires the addition of a period in a sentence that ends with a set-out equation. But I find extra dots in such equations confusing, so I don't use punctuation in set-out formulas, even if proper grammar says they should be there

I doubt that these books will have the four editions that Szegő did, but it seems likely there will be later editions. That means I especially welcome comments, corrections, missing topics and references, and information on new papers. For the latter, I much prefer a link to an online archive rather than have you send me attachments.

14.4. Review by: Peter Larkin Duren.
Mathematical Reviews MR2105088 (2006a:42002b).

See review 13.3 above.

15.1. From the Publisher.

The main theme of this book is the 'path integral technique' and its applications to constructive methods of quantum physics. The central topic is probabilistic foundations of the Feynman-Kac formula. Starting with main examples of Gaussian processes (the Brownian motion, the oscillatory process, and the Brownian bridge), the author presents four different proofs of the Feynman-Kac formula. Also included is a simple exposition of stochastic Ito calculus and its applications, in particular to the Hamiltonian of a particle in a magnetic field (the Feynman-Kac-Ito formula). Among other topics discussed are the probabilistic approach to the bound of the number of ground states of correlation inequalities (the Birman-Schwinger principle, Lieb's formula, etc.), the calculation of asymptotics for functional integrals of Laplace type (the theory of Donsker-Varadhan) and applications, scattering theory, the theory of crushed ice, and the Wiener sausage. Written with great care and containing many highly illuminating examples, this classic book is highly recommended to anyone interested in applications of functional integration to quantum physics. It can also serve as a textbook for a course in functional integration.

15.2. From the Preface.

Ever since the first edition went out of print, I have been approached by people asking how to get a copy. So I'm glad that the American Mathematical Society agreed to do this reprinting in the Chelsea series.

The material has worn well, and if I were writing about this today, I wouldn't change much. There have been many developments over the past thirty years after the index, and I've added a brief bibliographic note on some of these developments. Comments from Dirk Hundertmark were useful in its preparation, and I'd like to thank him for his help

I close with a story relevant to the second edition. In 1981, not long after the first edition appeared, I visited Moscow. A high point of my visit was an evening with Israel Gel'fand in Yaglom's apartment. Midway through the evening, Gel'fand leaned over and, with a twinkle in his eye, said to my wife, Martha: "Tell your husband his books are too good." I asked what he meant and he replied: "My son borrowed your Functional Integration book and he won't give it back."

Gel'fand senior, of course, accomplished his purpose. As soon as I got home, I sent him another copy of the book! But without realising it, it also served my purpose. His son, Sergei Gel'fand, is now a book acquisition editor for the American Mathematical Society! Not only has he been enthusiastic about this republication, but one of the two original copies sacrificed in the production of the second edition was provided by him. No doubt the copy mentioned to me in Moscow

16.1. From the Preface.

In 1977, I gave some lectures at the University of Texas which described the general theory of trace ideals initiated by von Neumann and Schatten (Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979). Because this theory has many different kinds of applications, the lecture notes I produced at the time were widely used, and I got many requests for information on how to obtain it once it fell out of print.

In 1993, I lectured at a summer school in Vancouver on the theory of applications of rank one perturbations of self-adjoint operators (Spectral analysis of rank one perturbations and applications in: Mathematical Quantum Theory, II. Schrödinger Operators, CRM Proc. Lecture Notes). The two topics have much in common. My interest in each arose in my research in several problems at once. And, of course, rank one perturbations are an extreme case of compact perturbations

Thus, when I started exploring the possibility of reprinting the Trace Ideals book as a second edition, it was natural to combine it with my Vancouver lectures. In preparing this new edition, I had to decide first whether to completely rewrite the material, and I chose not to because the basic theory hasn't changed much. Once I made that decision, I felt it made sense to only lightly edit the material, so that, for example, references to theorems or equation numbers would be the same. I fixed typos and made a few references to the addendum, especially where a conjecture had been settled. But I followed the original texts so closely that the notation in Chapters 1-10 (the original Texas lectures) and Chapters 11-14 (the Vancouver lectures) are, in a few points, slightly different.

I did add a much better index and an addendum describing some developments since the original notes were written

16.2. Review by: Pavel B Kurasov.
Mathematical Reviews MR2154153 (2006f:47086).

The first edition of this book has become a standard reference title in modern mathematical physics. It is highly appreciated due to its clarity and the excellent selection of its topics. Therefore the second edition of the book was long expected.

The book, devoted to trace ideals and their applications in mathematical physics, gives an excellent introduction to the subject as well as numerous examples. In particular, the following topics were described in the first edition: Calkin theory, traces and determinants, Fredholm theory, Kuroda-Birman scattering theory, bound state estimates, renormalisation in quantum field theory, and finally operator theory in Banach spaces.

For the second edition of the book additional chapters have been included. They are connected with the research of the author in the field of perturbation theory and were presented originally at the Vancouver Summer School in 1993. These lecture notes have served as a source of inspiration for several mathematical physicists since then and their publication in book form is highly appreciated. The first new chapter describes the general set-up for the theory of rank-one perturbations based on the Aronszajn-Donoghue theory. Connections with Krein's spectral shift function and the change of boundary conditions seen as a generalised rank-one perturbation are discussed. The second chapter describes the spectral theory of rank-one perturbations, in particular the invariance of the absolutely continuous spectrum and instability of the point spectrum. The last two chapters contain the proof of localisation in the Anderson model and an application of the xi function to the inverse spectral problem for one-dimensional Schrödinger operators and Jacobi matrices.

The book under review can be recommended to everybody working in mathematics and mathematical physics, from graduate students to active researchers. The book is also very well suited for a graduate course since the exposition is clear and perfectly self-contained.

17.1. From the Publisher.

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasises necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line.

In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC.

17.2. From the Preface.

In September 2006, I gave the Milton Brockett Porter Lectures at Rice University. Broadly speaking, these are the notes from those lectures. Of course, the five hours of lectures did not cover 600 pages of material. Roughly speaking, Chapters 1, 2, 3, 5, and 4 + 8 covered the topics of the five lectures, and I planned all along to include the material in Chapters 6 and 10, but Chapters 7 and 9 cover developments subsequent to Fall 2006.

The core motivation for the notes was to expose the developments of sum rule techniques that grew out of my paper with Rowan Killip. Another motivation was a lure David Damanik used to convince me to produce notes. He pointed out the irony that while the theory of periodic orthogonal polynomials was developed in the real line case (OPRL), there was a comprehensive presentation only for the unit circle case (OPUC), and suggested that I could include the OPRL case in these notes. It was in this regard that I "threw in" the chapter on Toda lattices (Chapter 6).

As mentioned, the inclusion of material that did not even exist in September 2006 added roughly 150 pages to the total. The biggest part concerns material on finite gap OPRL (Chapter 9). I intended the capstone of the lectures to be the work that Damanik, Killip, and I were then writing up on perturbations of periodic OPUC using sum rules. ...

But not all such finite gap sets arise in this way. ...
...
The reader may wonder why, given my earlier two-volume opus, there is a need for the current book and whether this book is not merely a subset of the earlier one. There is indeed some overlap, especially heavy in Chapters 2 and 3 of this book, but the overwhelming bulk of the material here is not covered in those earlier books. This current book has a rather different focus, concentrating on topics connected with sum rules and also on the real-line case rather than the unit circle. But there is another factor. It has been five years since I finished the other books and there have been some remarkable developments during this period. I have in mind not only some of my own work on finite gap systems (which are not OPUC in any event) but especially the beautiful work of Lubinsky on (Christoffel-Darboux) CD kernels and Remling on the a.c. spectrum, both of which are presented here. It is also true that in that period I have learned some things about OPUC that existed when my two-volume opus was written but were not included - notable here are the Helson-Lowdenslager approach to the singular component of the measure in the classical Szegő setup and the Máté-Nevai-Totik work on asymptotics of the diagonal CD kernel.

I would emphasise that while I have endeavoured to be accurate in the historical notes, I am not a serious historian of science, and that these notes are influenced by where I learned of some results and the generally accepted beliefs. When I was a graduate student, I took a course in the quantum theory of solids with Eugene Wigner. When he got to the discussion of the Brillouin zone, he remarked that they actually appeared first in the work of Bloch. When a student asked why it was called the Brillouin zone and not the Bloch zone, Wigner's face turned red and he stammered out: "Well, I learned of them from Brillouin's paper and named them, and the name stuck." The notes are here because they make the material more interesting and provide further references, rather than because they are intended as definitive history.

17.3. Review by: Harry Dym.
Mathematical Reviews MR2743058 (2012b:47080).

This book is a worthy companion to the author's earlier two-volume treatise on OPUC's, orthogonal polynomials on the unit circle. In this new work, centre stage is occupied by OPRL's, orthogonal polynomials on the real line R with respect to a measure d(x) and Jacobi matrices. In this book Jacobi matrices are real tridiagonal symmetric matrices (both finite and infinite) with strictly positive off-diagonal entries.
...
This book is a magnificent compilation of results of which the author can be justifiably proud. It can be used on many levels, running from a reference book for experts to a manual for beginners on what one really needs to know to pursue research in spectral theory. The latter objective is enhanced by the inclusion of tutorial sections on a variety of prerequisites, which also makes the individual chapters suitable for seminars and reading projects. It is also an excellent source of examples.

17.4. Review by: Christian Berg.
Journal of Approximation Theory.

Simon can write books faster than most people can read them. The quality is very high and the level of scholarship is enormous. ... The book is recommended to everyone who wants to broaden his or her knowledge about recent developments in orthogonal polynomials. It is a pleasure to see that many areas of mathematics are tied together.

17.5. Review by: Vladimir L Makarov.
Zentralblatt MATH

This book can be very useful for mathematicians who are familiar with the fields of analysis such as measure theory, orthogonal polynomials, and application of the theory of matrices and bounded linear operators.

17.6. Review by: Andrei Martínez-Finkelshtein.
University of Almería, Spain.

Simon is a leading specialist in orthogonal polynomials and spectral theory, with a very wide mathematical and physical background. This book contains a huge amount of new material found only in research papers. Those interested in orthogonal polynomials will find here many new results and techniques, while specialists in spectral theory will discover deep connections with topics from classical analysis and other areas.

17.7. Review by: Doron Lubinsky.
Georgia Institute of Technology.

This book is an important one. Barry Simon has revolutionised the study of orthogonal polynomials since he entered the field. Many of the fundamental advances he and his students pioneered appear here in book form for the first time. There is no question of his profound scholarship and expertise on this topic.

18.1. From the Publisher.

Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions including the infinite-dimensional case and emphasising the analytic point of view. Chapter one introduces the reader to the basic definitions and ideas that play central roles throughout the book. The rest of the book is divided into four parts: convexity and topology on infinite-dimensional spaces; Loewner's theorem; extreme points of convex sets and related issues, including the Krein-Milman theorem and Choquet theory; and a discussion of convexity and inequalities. The connections between disparate topics are clearly explained, giving the reader a thorough understanding of how convexity is useful as an analytic tool. A final chapter overviews the subject's history and explores further some of the themes mentioned earlier. This is an excellent resource for anyone interested in this central topic.

18.2. From the Preface.

Convexity of sets and functions are extremely simple notions to define, so it may be somewhat surprising the depth and breadth of ideas that these notions give rise to. It turns out that convexity is central to a vast number of applied areas, including Statistical Mechanics, Thermodynamics, Mathematical Economics, and Statistics, and that many inequalities, including Hölder's and Minkowski's inequalities, are related to convexity.

An introductory chapter (1) includes a study of the regularity properties of convex functions, some inequalities (Hölder's, Minkowski's, and Jensen's), the Hahn-Banach theorem as a statement about extending tangents to convex functions, and the introduction of two constructions that will play major roles later in this book: the Minkowski gauge of a convex set and the Legendre transform of a function

The remainder of the book is roughly in four parts: convexity and topology on infinite-dimensional spaces (Chapters 2-5); Loewner's theorem (Chapters 6-7); extreme points of convex sets and related issues, including the Krein-Milman theorem and Choquet theory (Chapters 8-11); and a discussion of convexity and inequalities (Chapters 12-16).

18.3. Table of Contents

Preface
1. Convex functions and sets
2. Orlicz spaces
3. Gauges and locally convex spaces
4. Separation theorems
5. Duality: dual topologies, bipolar sets, and Legendre transforms
6. Monotone and convex matrix functions
7. Loewner's theorem: a first proof
8. Extreme points and the Krein-Milman theorem
9. The strong Krein-Milman theorem
10. Choquet theory: existence
11. Choquet theory: uniqueness
12. Complex interpolation
13. The Brunn-Minkowski inequalities and log concave functions
14. Rearrangement inequalities: a) Brascamp-Lieb-Luttinger inequalities
15. Rearrangement inequalities: b) Majorization
16. The relative entropy
17. Notes

18.4. Review by: Semën Samsonovich Kutateladze.
Mathematical Reviews MR2814377 (2012d:46002).

Convexity is the central idea of functional analysis, and so this book turns around the analytical aspects of convexity. The latter includes conjugation, duality, and integral representation, all combined with inequality and extremality. All these topics are addressed in an attractive, lucid, and mostly definitive manner.

The book consists of 17 chapters of which the last contains intriguing and informative historical notes.
...
Simon's monograph is a valuable addition to the literature on convexity that will inspire many minds enchanted by the beauty and power of the cornerstone of functional analysis.

19.1. From the Publisher.

A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis.

Part 1 is devoted to real analysis. From one point of view, it presents the infinitesimal calculus of the twentieth century with the ultimate integral calculus (measure theory) and the ultimate differential calculus (distribution theory). From another, it shows the triumph of abstract spaces: topological spaces, Banach and Hilbert spaces, measure spaces, Riesz spaces, Polish spaces, locally convex spaces, Fréchet spaces, Schwartz space, and spaces. Finally it is the study of big techniques, including the Fourier series and transform, dual spaces, the Baire category, fixed point theorems, probability ideas, and Hausdorff dimension. Applications include the constructions of nowhere differentiable functions, Brownian motion, space-filling curves, solutions of the moment problem, Haar measure, and equilibrium measures in potential theory.

19.2. From the Preface for the five book series.

Reed-Simon starts with "Mathematics has its roots in numerology, geometry, and physics." This puts into context the division of mathematics into algebra, geometry/topology, and analysis. There are, of course, other areas of mathematics, and a division between parts of mathematics can be artificial. But almost universally, we require our graduate students to take courses in these three areas.

This five-volume series began and, to some extent, remains a set of texts for a basic graduate analysis course. In part, it reflects Caltech's three-terms-per-year schedule and the actual courses I've taught in the past. Much of the contents of Parts 1 and 2 (Part 2 is in two volumes, Part 2A and Part 2B) are common to virtually all such courses: point set topology, measure spaces, Hilbert and Banach spaces, distribution theory, and the Fourier transform, complex analysis including the Riemann mapping and Hadamard product theorems Parts 3 and 4 are made up of material that you'll find in some, but not all, courses: on the one hand, Part 3 on maximal functions and HP-spaces; on the other hand, Part 4 on the spectral theorem for bounded self-adjoint operators on a Hilbert space and det and trace, again for Hilbert space operators. Parts 3 and 4 reflect the two halves of the third term of Caltech's course.

While there is, of course, overlap between these books and other texts, there are some places where we differ, at least from many:

(a) By having a unified approach to both real and complex analysis, we are able to use notions like contour integrals as Stieltjes' integrals that cross the barrier.

(b) We include some topics that are not standard, although I am surprising they are not. For example, while discussing maximal functions, I present Garcia's proof of the maximal (and so, Birkhoff's) ergodic theorem

(c) These books are written to be keepers. The idea is that, for many students, this may be the last analysis course they take, so I've tried to write in a way that will be useful as a reference. For this reason, I've included "bonus" chapters and sections—material that I do not expect to be included in the course. This has several advantages. First, in a slightly longer course, the instructor has an option of extra topics to include. Second, there is some flexibility for an instructor who can't imagine a complex analysis course without a proof of the prime number theorem; it is possible to replace all or part of the (non-bonus) chapter on elliptic functions with the last four sections of the bonus chapter on analytic number theory. Third, it is certainly possible to take all the material in, say, Part 2, and turn it into a two-term course. Most importantly, the bonus material is there for the reader to peruse long after the formal course is over

(d) I have long collected "best" proofs and over the years learned a number of ones that are not the standard textbook proofs. In this regard, modern technology has been a boon. Thanks to Google Books and the Caltech library, I've been able to discover some proofs that I hadn't learned before. Examples of things that I'm especially fond of are Bernstein polynomials for the classical Weierstrass approximation theorem, von Neumann's proof of the Lebesgue decomposition and Radon-Nikodym theorems, the Hermite expansion treatment of the Fourier transform, Landau's proof of the Hadamard factorisation theorem, Wielandt's theorem on the functional equation for $F(z)$, and Newman's proof of the prime number theorem. Each of these appears in at least some monographs, but they are not nearly as widespread as they deserve to be

(e) I've tried to distinguish between central results and interesting asides and to indicate when an interesting aside is going to come up again later. In particular, all chapters, except those on preliminaries, have a listing of "Big Notions and Theorems" at their start. I wish that this attempt to differentiate between the essential and the less essential didn't make this book different, but alas, too many texts are monotone listings of theorems and proofs.

(f) I've included copious "Notes and Historical Remarks" at the end of each section. These notes illuminate and extend, and they (and the Problems) allow us to cover more material than would otherwise be possible. The history is there to enliven the discussion and to emphasise to students that mathematicians are real people and that "may you live in interesting times" is truly a curse. Any discussion of the history of real analysis is depressing because of the number of lives ended by the Nazis. Any discussion of nineteenth-century mathematics makes one appreciate medical progress, contemplating Abel, Riemann, and Stieltjes. I feel knowing that Picard was Hermite's son-in-law spices up the study of his theorem

On the subject of history, there are three cautions. First, I am not a professional historian, and almost none of the history discussed here is based on original sources. I have relied on information on the Internet. I have tried for accuracy, but I'm sure there are errors, some that would make a real historian wince.

A second caution concerns looking at the history assuming the mathematics we now know. Especially when concepts are new, they may be poorly understood or viewed from a perspective quite different from the one here. Looking at the wonderful history of nineteenth-century complex analysis by Bottazzini-Grey will illustrate this more clearly than these brief notes can.

The third caution concerns naming theorems. Here, the reader needs to bear in mind Arnol'd's principle: If a notion bears a personal name, then that name is not the name of the discoverer (and the related Berry principle: The Arnol'd principle is applicable to itself).
...
These books have a wide variety of problems, along with a multiplicity of uses. The serious reader should at least skim them since there is often interesting supplementary material covered there.

Similarly, these books have a much larger bibliography than is standard, partly because of the historical references (many of which are available online and a pleasure to read) and partly because the Notes introduce lots of peripheral topics and places for further reading. But the reader shouldn't consider for a moment that these are intended to be comprehensive - that would be impossible in a subject as broad as that considered in these volumes.

These books differ from many modern texts by focusing a little more on special functions than is standard. For much of the nineteenth century, the theory of special functions was considered a central pillar of analysis. They are now out of favour too much, although one can see some signs of the pendulum swinging back. They are still mainly peripheral but appear often in Part 2 and a few times in Parts 1, 3, and 4

These books are intended for a second course in analysis, but in most places, it is really previous exposure being helpful rather than required. Beyond the basic calculus, the one topic that the reader is expected to have seen is metric space theory and the construction of the reals as completion of the rationals (or by some other means, such as Dedekind cuts).
...
Finally, analysis is a wonderful and beautiful subject. I hope the reader has as much fun using these books as I had writing them.

19.3. From the Preface to Part 1.

I warn you in advance that all the principles ... that I'll now tell you about, are a little false. Counterexamples can be found to each one - but as directional guides the principles still serve a useful purpose. - Paul Halmos

Analysis is the infinitesimal calculus written large. Calculus as taught to most high school students and college freshmen is the subject as it existed around 1750 - I've no doubt that Euler could have gotten a perfect score on the Calculus BC advanced placement exam. Even "rigorous" calculus courses that talk about - proofs and the intermediate value theorem only bring the subject up to about 1890 after the impact of Cauchy and Weierstrass on real variable calculus was felt.

This volume can be thought of as the infinitesimal calculus of the twentieth century. From that point of view, the key chapters are Chapter 4, which covers measure theory - the consummate integral calculus and the first part of Chapter 6 on distribution theory - the ultimate differential calculus

But from another point of view, this volume is about the triumph of abstraction. Abstraction is such a central part of modern mathematics that one forgets that it wasn't until Fréchet's 1906 thesis that sets of points with no prior underlying structure (not assumed points in or functions on Rn) are considered and given a posteriori structure (Fréchet first defined abstract metric spaces). And after its success in analysis, abstraction took over significant parts of algebra, geometry, topology, and logic.

Abstract spaces are a distinct thread here, starting with topological spaces in Chapter 2, Banach spaces in Chapter 5 (and its special case, Hilbert spaces, in Chapter 3), and locally convex spaces in the later parts of Chapters 5 and 6 and in Chapter 9.

Of course, abstract spaces occur to set up the language we need for measure theory (which we do initially on compact Hausdorff spaces and where we use Banach lattices as a tool) and for distributions, which are defined as duals of some locally convex spaces.

Besides the main threads of measure theory, distributions, and abstract spaces, several leitmotifs can be seen: Fourier analysis (Sections 3.5, 6.2, and 6.4-6.6 are a minicourse), probability (Bonus Chapter 7 has the basics, but it is implicit in much of the basic measure theory), convexity (a key notion in Chapter 5), and at least bits and pieces of the theory of ordinary and partial differential equations

The role of pivotal figures in real analysis is somewhat different from complex analysis, where three figures - Cauchy, Riemann, and Weierstrass - dominated not only in introducing the key concepts, but many of the most significant theorems. Of course, Lebesgue and Schwartz invented measure theory and distributions, respectively, but after ten years, Lebesgue moved on mainly to pedagogy, and Hörmander did much more to cement the theory of distributions than Schwartz. On the abstract side, F Riesz was a key figure for the 30 years following 1906, with important results well into his fifties, but he doesn't rise to the dominance of the complex analytic three.

In understanding one part of the rather distinct tone of some of this volume, the reader needs to bear in mind "Simon's three kvetches":

1. Every interesting topological space is a metric space.

2. Every interesting Banach space is separable

3. Every interesting real-valued function is Baire/Borel measurable

Of course, the principles are well-described by the Halmos quote at the start they aren't completely true but capture important ideas for the reader to bear in mind. As a mathematician, I cringe at using the phrase "not completely true." I was in a seminar whose audience included Ed Nelson, one of my teachers. When the speaker said the proof he was giving was almost rigorous, Ed said: "To say something is almost rigorous makes as much sense as saying a woman is almost pregnant." On the other hand, Neils Bohr, the founding father of quantum mechanics, said: "It is the hall-mark of any deep truth that its negation is also a deep truth."

19.4. Review by: Fritz Gesztesy.
Mathematical Reviews MR3408971.

Where does one even begin with a review of this monumental treatise on analysis written by Barry Simon? The four parts (with part two on complex analysis being presented in two volumes, part 2A and part 2B) comprise a total of 3259 pages. This project differs in a variety of ways from traditional books on analysis, even if one compares it with those that cover a wide range of topics from real analysis all the way to operator theory. The presentation is encyclopaedic in nature and succeeds wonderfully in its attempt to provide a reference set. The bonus sections and chapters, which might be omitted in a standard course, represent enlightening additional material, suitable for a second reading or when one has the luxury of a multi-semester course. They also present the instructor with some flexibility to pick and choose certain sections.

Simon's collection of "best proofs" of fundamental theorems is on display in these volumes, as is a distinction between central results and important asides. The reader also benefits from his unique perspective as a mathematical physicist and analyst. Each chapter starts with a listing of Big Notions and Theorems to avoid the all too common monotonous litany of theorems and proofs in standard analysis texts. Moreover, extensive notes, historical remarks, and capsule biographies at the end of each section illuminate, extend, and complement the presentation. Together with the set of problems, these notes permit a substantial broadening of the scope of the entire project. In addition, a much larger than typical bibliography, including historical references, sets these volumes apart from others in this area.

Another interesting feature distinguishing these volumes from alternative contemporary treatises of analysis is that the apparently lost art of special functions is not only not shunned, but, as we will see below, is actually discussed in a refreshingly explicit manner, countering a regrettable trend in modern analysis.

This project represents a unique second course in analysis that will serve as a reference set and encyclopaedia for students and professionals alike. Students exposed to this project early on will have the distinct advantage of starting their advanced studies at an incredibly elevated level and, after a first reading, being able to turn back to it time and time again in their subsequent studies and future careers.

Part 1, Real analysis: The centrepiece of modern real analysis, measure theory, starting with positive functionals on the set of continuous functions on a compact Hausdorff space X, an approach pioneered by P Lax, is presented in Chapter 4 of Part 1 and so sets this volume apart from many real analysis treatises.
...
There is no doubt that graduate students and seasoned analysts alike will find a wealth of material in this project and appreciate its encyclopaedic nature.

20.1. From the Publisher.

Cauchy Integral Theorem, Consequences of the Cauchy Integral Theorem (including holomorphic iff analytic, Local Behaviour, Phragmén-Lindelöf, Reflection Principle, Calculation of Integrals), Montel, Vitali, and Hurwitz's Theorems, Fractional Linear Transformations, Conformal Maps, Zeros and Product Formulae, Elliptic Functions, Global Analytic Functions, and Picard's Theorem.

Selected topics include the Goursat Argument, Ultimate and Ultra Cauchy Integral Formulas, Runge's Theorem, complex interpolation, Marty's Theorem, continued fraction analysis of real numbers, Riemann mapping theorem, Uniformization theorem (modulo results from Part 3), Mittag-Leffler and Weirstrass product theorems, finite order and Hadamard product formula, Gamma function, Euler-Maclaurin Series and Stirling's formula to all orders, Jensen's formula and Blaschke products, Weierstrass and Jacobi elliptic functions, Jacobi theta functions, Paley-Wiener theorems, Hartog's phenomenon, and Poincaré's theorem that in higher complex dimensions, the ball and polydisk are not conformally equivalent.

20.2. From the Preface for the five book series.

See 19.2. above.

20.3. From the Preface to Part 2.

Part 2 of this five-volume series is devoted to complex analysis. We've split Part 2 into two pieces (Part 2A and Part 2B), partly because of the total length of the current material, but also because of the fact that we've left out several topics, so Part 2B has some room for expansion. To indicate the view that these two volumes are two halves of one part, the chapter numbers are cumulative. Chapters 1-11 are in Part 2A, and Part 2B starts with Chapter 12.

The flavour of Part 2 is quite different from Part 1 - abstract spaces are less central (although hardly absent) - the content is more classical and more geometrical. The classical flavour is understandable. Most of the material in this part dates from 1820-1895, while Parts 1, 3, and 4 largely date from 1885-1940

While real analysis has important figures, especially F Riesz, it is hard to single out a small number of "fathers." On the other hand, it is clear that the founding fathers of complex analysis are Cauchy, Weierstrass, and Riemann. It is useful to associate each of these three with separate threads which weave together into the amazing tapestry of this volume. While useful, it is a bit of an exaggeration in that one can identify some of the other threads in the work of each of them. That said, they clearly do have distinct focuses, and it is useful to separate the three points of view.

To Cauchy, the central aspect is the differential and integral calculus of complex-valued functions of a complex variable. Here the fundamentals are the Cauchy integral theorem and the Cauchy integral formula. These are the basics behind Chapters 2-5.

For Weierstrass, sums and products, and especially the power series, are the central object. These appear first peeking through in the Cauchy chapters (especially Section 2.3) and dominate in Chapters 6, 9, 10, and parts of Chapter 11, Chapter 13, and Chapter 14.

For Riemann, it is the view as conformal maps and associated geometry. The central chapters for this are Chapters 7, 8, and 12, but also parts of Chapters 10 and 11.

In fact, these three strands recur all over and are interrelated, but it is useful to bear in mind the three points of view.

20.4. Review by: Fritz Gesztesy.
Mathematical Reviews MR3443339.

Part 2A, Basic complex analysis: Part 2 focuses on the classical and geometric flavour of the subject created by the giants, Cauchy, Weierstrass, and Riemann, and their distinct points of view. Familiar topics like boundary values of analytic functions in the unit disk, as well as aspects of potential theory, are deferred to Part 3 due to their intimate connections with harmonic analysis.
...
There is no doubt that graduate students and seasoned analysts alike will find a wealth of material in this project and appreciate its encyclopaedic nature.

21.1. From the Publisher.

Conformal metric methods, topics in analytic number theory, Fuchsian ODEs and associated special functions, asymptotic methods, univalent functions, and Nevanlinna theory.

Selected topics include the Poincaré metric, Ahlfors-Robinson proof of Picard's theorem, Bergmann kernel, Painlevé's conformal mapping theorem, Jacobi 2- and 4-squares theorems, Dirichlet series, Dirichlet's prime progression theorem, zeta function, prime number theorem, hypergeometric, Bessel and Airy functions, Hankel and Sommerfeld contours, Laplace's method, stationary phase, steepest descent, WKB, Koebe function, Loewner evolution and introduction to SLE, and Nevanlinna's First and Second Main theorems.

21.2. From the Preface for the five book series.

See 19.2. above

21.3. From the Preface to Part 2.

See 20.3 above.

21.4. Review by: Fritz Gesztesy.
Mathematical Reviews MR3364090.

Part 2B, Advanced complex analysis: This volume seamlessly continues Part 2A and treats Riemannian metrics and complex analysis in Chapter 12. Various topics in analytic number theory, including Dirichlet series, the Riemann zeta- and Dirichlet L-function, and the prime number theorem, are treated in Chapter 13. Ordinary differential equations in the complex domain as well as hypergeometric, Bessel, and Airy functions are discussed in Chapter 14. Asymptotic methods, including Laplace's method, Watson's lemma, and the methods of stationary phase and steepest descent, are presented in Chapter 15. Univalent functions and Loewner evolution, and the basics of Nevanlinna theory are treated in Chapters 16 and 17; they complete Part 2B and hence this comprehensive account of complex analysis.
...
There is no doubt that graduate students and seasoned analysts alike will find a wealth of material in this project and appreciate its encyclopaedic nature.

22.1. From the Publisher.

Maximal functions and pointwise limits, harmonic functions and potential theory, phase space analysis, $Hp$ spaces, and more inequalities.

Selected topics include the Hardy-Littlewood maximal function, von Neumann and Birkhoff ergodic theorems, Weyl equidistribution, ergodicity of the Gauss (continued fraction) map, ergodicity of geodesic flow on certain Riemann surfaces, Kingman subadditive ergodic theorem, Ruelle-Oseledec theorem, martingale convergence theorem, subharmonic functions, Perron's method, spherical harmonics, Frostman's theorem, Kellogg-Evans theorem, potential theory on Riemann surfaces, pseudo-differential operators, coherent states, wavelets, BMO, real interpolation and Marcinkiewicz theorem, Hardy-Littlewood-Sobolev inequalities, Sobolev spaces, Calderón-Zygmund method, Calderón-Vaillancourt estimates, Hypercontractive and Log-Sobolev estimates, Lieb Thirring and CLR bounds, and the Tomas-Stein theorem.

22.2. From the Preface for the five book series.

See 19.2. above

22.3. From the Preface to Part 3.

I don't have a succinct definition of harmonic analysis, or perhaps I have too many. One possibility is that harmonic analysis is what harmonic analysts do. There is an active group of mathematicians, many of them students of, or grandstudents of, Calderón or Zygmund, who have come to be called harmonic analysts, and much of this volume concerns their work or the precursors to that work. One problem with this definition is that, in recent years, this group has branched out to cover certain parts of nonlinear PDEs and combinatorial number theory.

Another approach to a definition is to associate harmonic analysis with "hard analysis," a term introduced by Hardy, who also used "soft analysis" as a pejorative for analysis as the study of abstract infinite-dimensional spaces. There is a dividing line between the use of abstraction, which dominated the analysis of the first half of the twentieth century, and analysis that relies more on inequalities, which regained control in the second half. There is some truth to the idea that Part 1 in this series of books is more on soft analysis and Part 3 on hard analysis, but, in the end, both parts have many elements of both abstraction and estimates.

Perhaps the best description of this part is that it should really be called "More Real Analysis." With the exception of Chapter 5 on HP-spaces, any chapter would fit with Part 1. Indeed, Chapter 4, which could be called "More Fourier Analysis," started out in Part 1 until I decided to move it here

The topics that should be in any graduate analysis course, and often are, are the results on Hardy-Littlewood maximal functions and the Lebesgue differentiation theorem in Chapter 2, the very basics of harmonic and subharmonic functions, something about Hp-spaces, and Sobolev inequalities.

The other topics are exceedingly useful but are less often in courses, including those at Caltech. Especially in light of Calderón's discovery of its essential equivalence to the Hardy-Littlewood theorem, the maximum ergodic theorem should be taught. And wavelets have earned a place, as well. In any event, there are lots of useful devices to add to our students' toolkits

22.4. Review by: Fritz Gesztesy.
Mathematical Reviews MR3410783.

Part 3, Harmonic analysis: Given that modern harmonic analysis defies a straightforward definition and comprises a variety of topics according to many mathematicians, Simon views Part 3 as "more real analysis" and then goes on to present a variety of tools and techniques (such as maximal functions, harmonic and subharmonic functions, $H^{p}$-spaces, BMO, etc.) which will satisfy typical practitioners in harmonic analysis.
...
There is no doubt that graduate students and seasoned analysts alike will find a wealth of material in this project and appreciate its encyclopaedic nature.

23.1. From the Publisher.

Eigenvalue Perturbation Theory, Operator Basics, Compact Operators, Orthogonal Polynomials, Spectral Theory, Banach Algebras, and Unbounded Self-Adjoint Operators.

Selected topics include analytic functional calculus, polar decomposition, Hilbert-Schmidt and Riesz-Schauder theorems, Ringrose structure theorems, trace ideals, trace and determinant, Lidskii's theorem, index theory for Fredholm operators, OPRL, OPUC, Bochner-Brenke theorem, Chebyshev polynomials, spectral measures, spectral multiplicity theory, trace class perturbations and Krein spectral shift, Gel'fand transform, Gel'fand-Naimark theorems, almost periodic functions, Gel'fand-Raikov and Peter-Weyl theorems, Fourier analysis on LCA groups, Wiener and Ingham tauberian theorems and the prime number theorem, Spectral and Stone's theorem for unbounded self-adjoint operators, von Neumann theory of self-adjoint extensions, quadratic forms, Birman-Krein-Vershik theory of self-adjoint extensions, Kato's inequality, Beurling-Deny theorems, moment problems, and the Birman-Schwinger principle.

23.2. From the Preface for the five book series.

See 19.2. above

23.3. From the Preface to Part 4.

The subject of this part is "operator theory." Unlike Parts 1 and 2, where there is general agreement about what we should expect graduate students to know, that is not true of this part.

Putting aside for now Chapters 4 and 6, which go beyond "operator theory" in a narrow sense, one can easily imagine a book titled Operator Theory having a little overlap with Chapters 2, 3, 5, and 7: almost all of that material studies Hilbert space operators. We do discuss in Chapter 2 the analytic functional calculus on general Banach spaces, and parts of our study of compact operators in Chapter 3 cover some basics and the Riesz-Schauder theory on general Banach spaces. We cover Fredholm operators and the Ringrose structure theory in normed spaces. But the thrust is definitely toward Hilbert space

Moreover, a book like Harmonic Analysis of Operators on Hilbert Space or any of several books with "non-self-adjoint" in their titles have little overlap with this volume. So, from our point of view, a more accurate title for this part might be Operator Theory - Mainly Self-Adjoint and/or Compact Operators on a Hilbert Space.

That said, much of the material concerning those other topics, undoubtedly important, doesn't belong in "what every mathematician should at least be exposed to in analysis." But, I believe the spectral theorem, at least for bounded operators, the notions of trace and determinant on a Hilbert space, and the basics of the Gelfand theory of commutative Banach spaces do be long on that list.

Before saying a little more about the detailed contents, I should mention that many books with a thrust similar to this book have the name "Functional Analysis." I still find it remarkable and a little strange that the parts of a graduate analysis course that deal with operator theory are often given this name (since functions are more central to real and complex analysis), but they are, even by me.

One change from the other parts in this series of five books is that in them all the material called "Preliminaries" is either from other parts of the series or from prior courses that the student is assumed to have had (e.g., linear algebra or the theory of metric spaces). Here, Chapter 1 includes a section on perturbation theory for eigenvalues ​​of finite matrices because it fits in with a review of linear algebra, not because we imagine many readers are familiar with it

Chapters 4 and 6 are here as material that I believe all students should see while learning analysis (at least the initial sections), but they are related to, though rather distinct from, "operator theory." Chapter 4 deals with a subject dear to my heart - orthogonal polynomials. It's officially here because the formal proof we give of the spectral theorem reduces it to the result for Jacobi matrices, which we treat by approximation theory for orthogonal polynomials (it should be emphasized that this is only one of seven proofs we sketch). I arranged this, in part, because I felt any first-year graduate student should know how to derive these from recurrence relations for orthogonal polynomials on the real line. We fill out the chapter with bonus sections on some fascinating aspects of the theory

Chapter 6 involves another subject that should be on the required list of any mathematician: the Gelfand theory of commutative Banach algebras. Again, there is a connection to the spectral theorem, justifying the chapter being placed here, but the in-depth look at applications of this theory, while undoubtedly part of a comprehensive look at analysis, doesn't fit very well under the rubric of operator theory

23.4. Review by: Fritz Gesztesy.
Mathematical Reviews MR3364494.

Part 4, Operator theory: The basics of bounded operator theory (and unbounded operator theory in the self-adjoint case) with particular attention to aspects of compact operators (predominantly in Hilbert spaces), comprising trace ideals, determinants, Fredholm operators and their index, fundamentals of orthogonal polynomials, and Banach algebras, constitute Part 4. As one of its centrepieces, the spectral theorem for bounded operators and its three canonical variants, resolutions of the identity, the functional calculus, and spectral measures, are presented in great depth.
...
There is no doubt that graduate students and seasoned analysts alike will find a wealth of material in this project and appreciate its encyclopaedic nature.

24.1. From the Publisher.

This book provides an in depth discussion of Loewner's theorem on the characterisation of matrix monotone functions. The author refers to the book as a 'love poem,' one that highlights a unique mix of algebra and analysis and touches on numerous methods and results. The book details many different topics from analysis, operator theory and algebra, such as divided differences, convexity, positive definiteness, integral representations of function classes, Pick interpolation, rational approximation, orthogonal polynomials, continued fractions, and more. Most applications of Loewner's theorem involve the easy half of the theorem. A great number of interesting techniques in analysis are the bases for a proof of the hard half. Centred on one theorem, eleven proofs are discussed, both for the study of their own approach to the proof and as a starting point for discussing a variety of tools in analysis. Historical background and inclusion of pictures of some of the main figures who have developed the subject, adds another depth of perspective.

The presentation is suitable for detailed study, for quick review or reference to the various methods that are presented. The book is also suitable for independent study. The volume will be of interest to research mathematicians, physicists, and graduate students working in matrix theory and approximation, as well as to analysts and mathematical physicists.

24.2. From the Preface.

This book is a love poem to Loewner's theorem. There are other mathematical love poems, although not many. One sign, not always present and not fool proof, is that like this book, the author has included pictures of some of the main figures in the development of the subject under discussion. The tell-tale sign is that the reader's initial reaction is "how can there a whole book on that subject" (although not all narrow books are love poems).

Loewner's theorem concerns the theory of monotone matrix functions, i.e. functions, $f$ so that $A ≤ B \rightarrow f(A) ≤ f(B)$ for pairs of selfadjoint matrices. That this is a subtle notion is seen by the fact (see Corollary 14.3) that if$f: R \rightarrow R$ is monotone on all pairs of 2 × 2 matrices, then $f$ is affine! So Loewner realised one needed to fix a proper interval $(a, b) \subset \mathbb{R}$, demand that $f: (a, b) \rightarrow \mathbb{R}$, and only demand the monotonicity result for pairs $A$ and $B$ all of whose eigenvalues are in $(a, b)$. In 1934, Charles Loewner proved the remarkable result that $f$ on $(a, b)$ is matrix monotone on all such $n \times n$ pairs (for all $n$) if and only if $f$ is real analytic on $(a, b)$ and has an analytic continuation to the upper half plane with a positive imaginary part there. That functions with this property are matrix monotone follows in a few lines from the Herglotz representation theorem, so I call that half the easy half. The other direction is the hard half. Many applications of Loewner's notion of matrix monotone functions involve explicit examples and so only the easy half.

Matrix monotonicity is an algebraic statement, but Loewner's theorem says it is equivalent to an analytic fact. One fascination of the subject is the mix of the algebraic and the analytic in its study.

Interestingly enough, this is not the first love poem to Loewner's theorem. Forty-five years ago, in 1974, Springer published W F Donoghue's Monotone Matrix Functions and Analytic Continuation in the same series where I am publishing this book. Donoghue explained that, at the time of writing, there were three existing proofs of (the hard half of) Loewner's theorem, all very different: Loewner's original proof, the proof of Bendat-Sherman, and the proof of Korányi. In fact, as we'll discuss several times in this book, there was a fourth proof which workers in the field didn't seem to realise was there it was due to Wigner-von Neumann and appeared in the Annals of Mathematics! The main goal of Donoghue's book was to expose those three proofs with their underlying mathematical background. He also had several new results including converses of two theorems of Loewner's student Otto Dobsch. Donoghue also discussed applications to the theory of schlicht and Herglotz functions.

Similarly, the main goal of the current book is to expose many of the proofs of the hard half that now exist. Indeed, we will give 11 proofs in all. As I'll explain later, there is a 12th proof which I don't expose. Of course, it is not always clear when two proofs are really different and when one proof should be regarded as a variant of another. I am prepared to defend the notion that these proofs are different (although there are relations among them) and that a proof like Hansen's should be viewed as a variant (albeit an interesting variant) of Hansen's earlier proof with Pedersen. But I'd agree some might differ.

In any event, it is striking that there are no really short proofs and that the underlying structure of the proofs is so different. I like to joke that it is almost as if every important result in analysis is the basis of some proof of Loewner's theorem. To mention a few of the underlying machines that lead to proofs: the moment problem, Pick's theorem, commutant lifting theorems, the Krein-Milman theorem, and Mellin transforms.

Shortly after Loewner's 1934 paper, his student Fritz Kraus defined and began the study of matrix convex functions. Donoghue includes Kraus' 1936 paper in his bibliography but never refers to it in the text and doesn't discuss the subject of matrix convex or concave functions. In the years since his book, it has become clear that the two subjects are intertwined in deep ways. So my book also has a lot of material on matrix convex functions.

I should say something about the history of this book. I first learned of Loewner's theorem in graduate school 50 years ago, probably from Ed Nelson, one of my mentors. I learned the proof of the easy half and a variant of it appears in Reed-Simon. The easy half seemed more useful since it told one certain function was matrix monotone and that was what could be applied. About 2000, I became curious how one proved the hard half and looked at the expositions in Donoghue and Bhatia's books. I was intrigued that both quoted a paper of Wigner-von Neumann claiming it had applications to quantum physics. I looked up the paper and was surprised to discover that there was no application to quantum physics but there was a complete and different proof of Loewner's theorem, which Loewner, Donoghue, and Bhatia didn't seem to realise was there! For several years, I gave a mathematics colloquium called "The Lost Proof of Loewner's Theorem." I found my new proof that appears in Chapter 20 which is, in many ways, the most direct proof. After one of the colloquia, around 2005, I learned of Boutet de Monvel's unpublished proof and started on this book. I finished about half and then put it aside in 2007 to focus on my five-volume Comprehensive Course. After completing that, I returned to this book and completed it.
...
There is some literature on multivariable extensions of Loewner's theorem. The earliest such papers are a series (Korányi, Singh-Vasudeva, and Hansen) ... Recently, there have been several papers (Agler et al., Najafi, Pascoe and Tully-Doyle, and Pálfia) that involve functions of several variables, even non-commuting variables, on a single Hilbert space. These papers are complicated, and the subject is in flux, so it seemed wisest not to discuss the subject in detail here. However, we should mention that Pálfia says that his multivariable proof specialised to one variable provides a new proof of the classical result, the 12th proof referred to above. In correspondence, he told me that the restriction to one variable doesn't especially simplify his proof (his preprint is 40 pages). He didn't believe it could be shortened to less than about 25 pages and that doesn't count any mathematical background on the tools he uses, so I decided not to try to include it.

The 11 proofs appear in Part II of this book which also has background on tools like Pick's theorem and rational approximation. There is also a discussion of the analogue of Loewner's theorem when $(a, b)$ is replaced by a more general open set. Part I sets the background for the theorem and also for the theory of matrix convex functions. Part III discusses some applications, many due to Ando. Each part begins with a brief introduction that summarises the content and context of that part.

Karel Löwner was born near Prague in 1893 and proved his great theorem while a professor at the Charles University in Prague. In 1938, he fled from there to the United States (he was arrested when the Nazis entered Prague probably because of his left-wing political activities rather than the fact that he was Jewish) where he made a decision to change his name to the less German Charles Loewner. I have decided to respect his choice by calling the result Loewner's theorem. The reader should be warned that much of the literature refers to Löwner's theorem. Of course, we use the original spelling in the bibliography when that is what appeared in the author's name or in the published title of an article or book.

24.3. Review by: John E McCarthy.
Bulletin of the American Mathematical Society 57 (4) (2020), 769-684.

Simon's book studies Loewner's theorem in one variable, and gives eleven different proofs of the hard direction. As he puts it, the book is a love poem to Loewner's theorem.

Part I of the book is called Tools, and here he introduces Nevanlinna's theorem, which he calls the Herglotz representation theorem (since it follows from a 1911 theorem of Herglotz for functions on the disk), and he uses it to prove the easy direction. He then proves part (i) of Loewner's theorem and follows with an extensive treatment of matrix convex functions.

Part II starts by giving 4 different proofs of Pick's theorem (and later a fifth), a theorem first proved by Pick in 1916. Pick's theorem describes when a function mapping the upper half-plane to itself and satisfying given interpolation conditions exists; part (ii) of Loewner's theorem is a boundary value version.
...
The book is very carefully written. For each proof, the author gives all the necessary background material and includes detailed historical remarks. It is an impressive tour-de-force.

I would consider this book a mathematics lounge book - the sort of book that should be left lying around the lounge, so that one can dip into it over coffee, read a section, and then go back to work. It is accessible and is a delight to read.

24.4. Review by: Linda J Patton.
Mathematical Reviews MR3969971.

The main part of the book contains eleven different proofs of the hard direction of Loewner's theorem, where one assumes a function is matrix monotone and must establish that it is analytic and maps the upper half-plane to itself. This section begins with a comprehensive review of Pick interpolation. One of the eleven proofs is a new one by the author; he uses the close relationship between Loewner matrices and Pick matrices to establish Loewner's theorem as a natural limiting case of upper half-plane Pick interpolation. Several chapters present rational interpolation as used in Loewner's original proof and others. The proof by Loewner, as well as proofs by Korányi and by Bendat and Sherman, was also in Donoghue's 1974 monograph. Additional proofs that were published after Donoghue's book include those by Hansen and Pedersen, Sparr, Ameur, and Rosenblum and Rovnyak. There is a 1954 proof by Wigner and von Neumann that was not well known in the mathematical community. The author adds another new derivation in which he simplifies Wigner and von Neumann's continued fraction method. A third new proof that uses Mellin transforms comes from previously unpublished notes of Boutet de Monvel. When presenting the older proofs, the author follows the spirit of the derivation but often has simplifying modifications. For example, he uses a different moment problem in the Bendat-Sherman proof. The history and nuances of each proof are discussed.

Some generalisations and applications are in the last part of the book. Loewner's theorem has recently been extended to functions of several variables, in some cases in the non-commuting setting, by Agler, McCarthy and Young, Najafi, Pascoe and Tully-Doyle, Pascoe, and Pálfia. The author notes that he excluded the multivariable results because they are so recent and would require extensive additional background material.

This book will be a valuable reference for anyone interested in any aspect of Loewner's theorem. The variety of techniques used in the eleven proofs also makes the text a good introduction to many standard methods in functional analysis and function theory. Most of the necessary basic results are proved in detail; any missing steps are clearly referenced, often to the author's recent set of analysis texts. In addition to multiple proofs of Loewner's theorem, many of the background results have several different proofs. The author calls this monograph "a love poem to Loewner's theorem". His enthusiasm for the subject matter and occasional mathematical excursions make reading the book feel like an animated conversation with a colleague or teacher.