Günter Lumer was born in Germany, received some of his early education in France (where he was Guy Lumer), and went to university in Uruguay (which was the permanent haven his parents found away from Hitler). He spoke both German and French like a native, and Spanish almost as well. He is a short man, with a hooked nose, a wide grin, and a vivacious manner; he is always in motion. His size used to vary from normal to nearly obese so often that he needed two complete wardrobes, the fat one and the thin one.

His positive personality was visible even in his mathematical attitudes, even when he was a young student. What he was interested in at the moment was the most important problem in the world, and what's more he had just solved it - well, almost solved it - there is, you see, just this minor point that remains - is there a curve in the plane such that ... ? - and, great generalisation, is there a surface in space such that ... ? When I met him he was (without being aware of it) a spiritual descendant of R L Moore. I took it to be one of my jobs to show him a window to the rest of the world of mathematics. I opened the window, I pointed, I argued - he closed his eyes, he resisted, he argued - and it all ended well. He was energetic and talented enough that it's not at all clear that I really did him any good; I am sure he would have become a mathematician no matter what I did.
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Lumer made a lot of small mistakes, kept correcting them, and converged to the truth by successive approximations. ... Lumer's stuff would come out structurally organised but chaotic in its details ... Lumer [went on to do] hard Hardy spaces in Belgium ...

If H is a complex Hilbert space and if A is an operator on H (i.e., a bounded linear transformation of H into itself), under what conditions does there exist an operator B on H such that $B^{2} = A$? In other words, when does an operator have a square root? The spectral theorem implies that the normality of A is a sufficient condition for the existence of B; the special case of positive definite operators can be treated by more elementary means and is, in fact, often used as a step in the proof of the spectral theorem. As far as we are aware, no useful necessary and sufficient conditions for the existence of a square root are known, even in the classical case of finite-dimensional Hilbert spaces. The problem of finding some easily applicable conditions is of interest, in part because the use of square roots is frequently a helpful technique in the study of algebraic properties of operators, and in part because of the information that such conditions might yield about the hitherto rather mysterious behaviour of non-normal operators.

It was recently shown that on an infinite-dimensional Hilbert space there exist invertible operators with no square roots. The main purpose of this paper is to show that there are many such operators. More precisely, we shall show that the operators with square roots are not even dense among the invertible operators, or, in other words, that there exists a nonempty open set consisting entirely of invertible operators with no square roots.

I wish to express my gratitude to Professor I Kaplansky, under whose guidance this thesis was written, for many valuable suggestions and discussions. (It was in connection with a concrete problem proposed by Professor Kaplansky that I conceived of the present work.) I am also grateful to Professor P R Halmos for his influence on my mathematical education and later work. Last, but not least, I wish to thank Professor I E Segal, Professor S Sternberg, and Dr A Koranyi for many helpful conversations.

The theory that we shall develop extends the classical notions of numerical range (of an operator) and states (on an algebra of operator) to the case of operators on any Banach space. This provides one with new methods of attacking certain problems regarding the algebra of operator on a Banach space, in which the structure of the underlying space comes in explicitly.

The Hilbert space theory of dissipative operators was motivated by the Cauchy problem for systems of hyperbolic partial differential equations, where a consideration of the energy of, say, an electromagnetic field leads to an $L_{2}$ measure as the natural norm for the wave equation. However there are many interesting initial value problems in the theory of partial differential equations whose natural setting is not a Hilbert space, but rather a Banach space. Thus for the heat equation the natural measure is the supremum of the temperature whereas in the case of the diffusion equation the natural measure is the total mass given by an $L_{1}$ norm. In the present paper a suitable extension of the theory of dissipative operators to arbitrary Banach spaces is initiated.

During the academic year 1967/68 Günter stayed at Strasbourg University, thus close to my home University Saarbrücken. In the summer term 1967 he gave a series of lectures in Saarbrücken, and in the winter term 1967/68, which I spent at Caltech in Pasadena, a little bus supplied by our University brought my students to his lectures in Strasbourg every week. In the academic year 1969/70 Günter Lumer together with Irving Glicksberg organised a Research Seminar on function algebras at their home University, the University of Washington in Seattle. I had the good fortune to participate for three months on his invitation.

His scientific activities greatly contributed to the standing of the Belgian Universities in general and the University of Mons-Hainaut in particular. In 1976, supported by the Belgium National Science Foundation, Günter founded a contact group with the goal of organising research and exchange meetings in the fields of Partial Differential Equations and Functional Analysis. From the 1990s on, building on the success of this group, Günter became a driving force and leading contributor to several large-scale projects sponsored by the European Community. The resulting conferences on Evolution Equations created a lasting network supporting international research collaboration. These activities, combined with Günter's relentless energy and love for mathematics, were at the origin of the breath-taking development of the field of evolution equations and the theory of operator semigroups after the pioneering book of Hille and Phillips from 1957. In particular, between 1992 and 1997 he co-organised the North West European Analysis Seminar that was held in 1992 at Saint Amand les Eaux (France), in 1993 at Schloss Dagstuhl (Germany), in 1994 at Noordwijkerhout (The Netherlands), in 1995 at Lyon (France), in 1996 at Glasgow (United Kingdom) and in 1997 at Blaubeuren (Germany). Those seminars covered a broad range of topics in analysis and were a reflection of the true spirit of Günter Lumer, who always enjoyed bringing together and working with a wide range of mathematicians and scientists.

... is named after Gunter Lumer, a great mathematician with extensive achievements and a deep curiosity for applied functional analysis and operator theory, and is a joint scientific activity of all Action's Working Groups. The lectures will be delivered by carefully selected distinguished scientists whose research interests are related to our Action.