# Günter Lumer

### Quick Info

Frankfurt an Main, Germany

Brussels, Belgium

**Günter Lumer**was born in Germany but his Jewish family were forced to flee from the Nazis in 1933, eventually going to Uruguay where Günter was educated. He spent half his career in the United States and half in Belgium. He was a creative and prolific mathematician whose works have great influence on the research community in functional analysis and evolution equations.

### Biography

**Günter Lumer**was the son of Max Lumer (born 15 April 1887) and Hedwig Hella (born 9 May 1891). He had a brother Walter Lumer. Max Lumer had an electrical wholesale business in Frankfurt an Main and he married Hedwig Hella in Frankfurt on 30 October 1922. The family were Jewish and this presented them with great difficulties in 1933. On 30 January 1933 Hitler came to power and on 7 April 1933 clause three of the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. This did not directly affect Max Lumer, but it was clear to him that bringing up and educating Jewish children in Germany would present great difficulties. He moved his family to France in 1933 but retained his electrical wholesale business in Frankfurt an Main where it remained listed until 1937.

In France, Günter Lumer was known as Guy Lumer and there he began his education. The biographies all show that the Lumer family fled from France in 1941 and emigrated to Uruguay to escape the Nazis after the fall of France. After the war ended, the Allies made an effort to document people persecuted by the Nazi regime. The American Zone in Germany recorded the German Jew Hans Lumer of Frankfurt an Main as one of those persecuted.

Lumer entered the University of Uruguay in Montevideo to study electrical engineering. This, of course, was a natural topic for Lumer, following the interests and expertise of his father. Paul Halmos held visiting appointments at the University of Montevideo, Uruguay in 1951-1952. Lumer, who studied mathematics as part of the electrical engineering course, attended lectures by Halmos who tried to persuade him that mathematics and not electrical engineering was the subject for him. Halmos writes in [8]:-

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.Lumer was not the only student that Halmos encouraged to undertake research in mathematics while at the University of Montevideo. The other student was Juan Jorge Schäffer who, like Lumer was studying engineering and taking mathematics courses given by Halmos. Halmos, Lumer and Schäffer wrote the joint paper

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.

...

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 ...

*Square roots of operators*which they submitted to the American Mathematical Society on 4 April 1952. The paper begins [9]:-

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.In 1954 Halmos and Lumer published a follow-up paper [10] which begins:-

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.Despite these excellent mathematics papers, Lumer was still studying electrical engineering at the University of Uruguay and he graduated in 1957 with an engineering degree. He had, however, already made the decision to change to mathematics and had been awarded a Guggenheim Fellowship in 1956 to enable him to undertake research for a Ph.D. in the United States. After his close working with Halmos, it is not surprising that Lumer went to the University of Chicago where Halmos was working. At Chicago, Lumer's doctoral studies were supervised by Irving Kaplansky and he submitted his thesis

*Numerical Range and States in the Theory of Banach Space Operators*in August 1959. He gives the following acknowledgement in the thesis:-

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 Introduction to the thesis begins:-

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.We should make it clear that Lumer had several papers published while he was an undergraduate in Montevideo in addition to those with Halmos, and also while he was in Chicago, such as:

*Fine structure and continuity of spectra in Banach algebras*(1954),

*Sets with connected spherical section*(Portuguese) (1955),

*Polygons inscriptible in convex curves*(Spanish) (1956), (with A Jones)

*A note on radical rings*(Spanish) (1956),

*The range of the exponential function*(1957),

*Commutators in Banach algebras*(Spanish) (1957), (with Marvin Rosenblum)

*Linear operator equations*(1959).

Lumer did not return to Uruguay but remained in the United States where he was appointed to a one-year position at the University of California at Los Angeles for 1959-60. After this, he had another one-year appointment, this time at Stanford University for the academic year 1960-61. On 18 May 1961 he married Linda Naim (1929-2009) in San Francisco, California. Linda, the daughter of Clement Naim and Aimee Mimoun, had been born on 11 February 1929 in Tripoli, Libya into a Jewish family. After their marriage, the couple spent a honeymoon in France, flying back from Paris to New York on 31 July 1961. Günter and Linda had three children, the eldest being a son Marc Lumer born on 11 November 1962. Marc became an author, illustrator and graphic designer.

After these two one-year appointments, Lumer was appointed to the University of Washington where he worked from 1961 to 1974. It was in 1961 that Lumer published the result which today bears his name, the Lumer-Phillips theorem, which gives necessary and sufficient conditions on an operator to generate a strongly continuous semigroup of contractions on a general Banach space. The result appears in the paper

*Dissipative operators in a Banach space*published in the

*Pacific Journal of Mathematics*by Lumer and Ralph Saul Phillips (1913-1998). Ralph Phillips was on the staff at the University of California at Los Angeles in the year that Lumer spent there and moved to Stanford University in 1960, spending a second year with Lumer. The introduction to the paper begins as follows:-

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.In 1962 Lumer attended, by invitation, the functional analysis conference in Oberwolfach, the research centre in the Black Forest in Germany. He spent the academic year 1967-68 at Strasbourg University. Heinz König writes in [11]:-

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.In 1973 Lumer moved from the United States to take up a permanent position in Europe at the University of Mons-Hainaut in Mons, Wallonia, Belgium. This university in the French community of Belgium had French as its official language. The university had only been founded a few years before, in 1965, and in 2009, five years after Lumer died, it became part of the University of Mons [12] (see also [14]):-

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.We note that [12] lists 109 papers published by Lumer, the last being

*A general "isotropic" Paley-Wiener theorem and some of its applications*published in 2003.

In 1997 a serious hip joint operation reduced Lumer's mobility but he was able to continue with most of his mathematical activities. In addition to his work at the University of Mons-Hainaut, from 1999 to 2005 he was on the staff at the International Solvay Institutes for Physics and Chemistry in Brussels. He continued working at both these until his death in 2005. His wife Linda outlived him by five years and died in Brussels on 19 April 2009.

The Lumer Lectures were set up by the European Cooperation in Science and Technology Action CA18232, Mathematical Models for Interacting Dynamics on Networks. The series of Lumer Lectures [13]:-

... 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.

### References (show)

- J Beery and C Mead, Günter Lumer. Who's That Mathematician? Paul R Halmos Collection,
*Mathematical Association of America*.

https://www.maa.org/press/periodicals/convergence/whos-that-mathematician-paul-r-halmos-collection-page-32 - Günter Lumer,
*John Simon Guggenheim Memorial Foundation*.

https://www.gf.org/fellows/all-fellows/gunter-lumer/ - Günter Lumer,
*ancestry.co.uk*. - Günter Lumer, in
*American Men & Women of Science. A biographical directory of today's leaders in physical, biological, and related sciences*(12th edition) (R R Bowker, New York, 1971-73). - Günter Lumer, in
*American Men & Women of Science. A biographical directory of today's leaders in physical, biological, and related sciences*(13th edition) (R R Bowker, New York, 1976). - Günter Lumer, in
*Who's Who in America*(38th edition) (Marquis Who's Who, Wilmette, IL, 1974). - Günter Lumer, in
*Who's Who in the World*(5th edition) (Marquis Who's Who, Wilmette, IL, 1980). - P R Halmos,
*I Want To Be A Mathematician*(Springer-Verlag, New York, 1985), 187-188. - P R Halmos, G Lumer and J J Schäffer, Square roots of operators,
*Proceedings of the American Mathematical Society***4**(1953), 142-149. - P R Halmos and G Lumer, Square roots of operators II,
*Proceedings of the American Mathematical Society***5**(1954), 589-595. - H König, In Remembrance of Günter Lumer, in
*H Amann, W Arendt, M Hieber, F Neubrander, S Nicaise and J von Below (eds), Functional Analysis and Evolution Equations. The Günter Lumer Volume*(Birkhäuser, Basel-Boston-Berlin - Life and Work of Günter Lumer, in
*H Amann, W Arendt, M Hieber, F Neubrander, S Nicaise and J von Below (eds), Functional Analysis and Evolution Equations. The Günter Lumer Volume*(Birkhäuser, Basel-Boston-Berlin, 2008) - Lumer Lectures,
*Mathematical Models for Interacting Dynamics on Networks.*

https://mat-dyn-net.eu/en/lumer - S Nicaise, Günter Lumer (1929-2005),
*Université de Mons*.

https://math.umons.ac.be/en/membres/Gunter_Lumer.html

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Written by J J O'Connor and E F Robertson

Last Update February 2023

Last Update February 2023