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Selecta I - Research Contributions

Paul R. Halmos
Publication Date: 
Number of Pages: 
Springer Collected Works in Mathematics
[Reviewed by
Fernando Q. Gouvêa
, on

The unique shape of Paul Halmos’s Selecta reflects his unusual role in the mathematics of his time. Usually, a mathematician’s selected papers are simply presented in chronological order. Whatever claim they may have on the attention of future mathematicians is contained in the new ideas and new theorems they contain. Halmos, by contrast, has arranged his selected works into two volumes, one containing his research papers and the other giving a selection of his expository writing. I suspect he knew that his more important contribution was the latter.

Halmos was, of course, a very good mathematician, and his papers on Ergodic Theory and Operator Theory are important; anyone in either field would learn much from reading them. It is as an expositor, editor, and author of books, however, that Halmos had the most impact.

He began early. Late in the 1930s, with a brand-new Ph.D. in hand, Halmos became John von Neumann’s assistant at the Institute for Advanced Study. His notes on von Neumann’s course became Halmos’s first book, Finite-Dimensional Vector Spaces. It is not really an exaggeration to say that with that book, von Neumann and Halmos created a new area of mathematics, which we now call Linear Algebra. The new subject brought together a wide range of known results and made them into a coherent whole. Halmos would go on to write many other important books, on Hilbert Spaces, Measure Theory, Ergodic Theory, and Set Theory.

The writing of expository articles came naturally, though one can see a shift as time goes by: the earliest ones are mostly close to Halmos’s research interests, but the topics get more general and the concerns broader after the 1970s. Many of the more expository papers from that period appeared in the American Mathematical Monthly. Some are straightforward career advice: “How to Write Mathematics”, “How to Talk Mathematics”, “What to Publish”. A few verge on the philosophical: “Does Mathematics Have Elements?”, “Mathematics as a Creative Art”, and “Applied Mathematics is Bad Mathematics”, which shows that Halmos enjoyed being provocative as well.

The Selecta was originally published in 1983, and is reprinted here unchanged. We read in Leonard Gillman’s preface to volume II, for example, that Halmos has just begun his term as editor of the American Mathematical Monthly. There was, of course, much to come; a search on MathSciNet for material published after that date finds several papers and books, including Halmos’s autobiography, I Want to be a Mathematician. Maybe there is not enough for a third volume, but these could have been expanded. 

Nevertheless, we should be grateful that Springer has continued to make its many volumes of collected works available in paperback. The organization of these volumes makes things easier for most potential readers: if you work in (or close to) Halmos’s research field, you might want to consider volume I; everyone should buy and read volume II.

Fernando Gouvêa hopes he is a mathematician.

Work in operator theory, by Donald Sarason
Work in Ergodic theory, by Nathaniel Friedman
[1939 a] Invariants of certain stochastic transformations: the mathematical theory of gambling systems
[1942 d] On monothetic groups
[1942 e] Operator methods in classical mechanics, II
[1943] On automorphisms of compact groups
[1944 a] Approximation theories for measure preserving transformations
[1944 d] In general a measure preserving transformation is mixing
[1947 c] Invariant measures
[1949 b] Application of the Radon-Nikodym theorem to the theory of sufficient statistics
[1950 a] Normal dilations and extensions of operators
[1950 b] Commutativity and spectral properties of normal operators
[1950 c] The marriage problem
[1952 a] Spectra and spectral manifolds
[1952 b] Commutators of operators
[1953 a] Square roots of operators
[1954 a] Commutators of operators, II
[1958 b] Products of symmetries
[1961 c] Shifts on Hilbert spaces
[1963 c] Partial isometries
[1963 d] Algebraic properties of Toeplitz operators
[1964 a] Numerical ranges and normal dilations
[1965 b] Cesàro operators
[1966] Invariant subspaces of polynomially compact operators
[1968 b] Irreducible operators
[1968 c] Quasitriangular operators
[1969 a] Invariant subspaces 1969
[1969 b] Two subspaces
[1970 d] Ten problems in Hilbert space
[1971 a] Capacity in Banach algebras
[1972 a] Continuous functions of Hermitian operators
[1972 b] Positive approximants of operators
[1972 c] Products of shifts
[1973 b] Limits of shifts
[1974 a] Spectral approximants of normal operators
[1976 c] Some unsolved problems of unknown depth about operators on Hilbert space
[1979 b] Ten years in Hilbert space
[1980 a] Limsups of Lats
[1982 d] Asymptotic Toeplitz operators.