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A Glimpse at Hilbert Space Operators

Sheldon Axler, Peter Rosenthal, and Donald Sarason, editors
Publication Date: 
Number of Pages: 
Operator Theory Advances and Applications 207
[Reviewed by
Gordon MacDonald
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This is a book about the legacy of Paul Halmos in Operator Theory — the people and the mathematics he inspired and influenced.

While Halmos did not consider himself to be one of the great research mathematicians, he was certainly a damn good one. He made significant contributions to a number of fields including Ergodic Theory, Measure Theory, and especially Operator Theory. He had a knack for asking the right questions. In his 1970 paper “Ten problems in Hilbert space” he posed and discussed what he considered to be important open problems in the subject. In every case, consideration of these problems led to significant mathematics well beyond the initial scope of the problem. For example, the first problem solved, “Is every normal operator of the form diagonal plus compact?”, led to consideration of essentially normal operators, which led to the Brown-Douglas-Fillmore classification, which was a seminal result in the whole field of K-theory for C*-algebras

If you are not an Operator Theorist and you know of Halmos, it is probably due to his work as an expositor and teacher of mathematics. He wrote 15 textbooks, including the well-known Naive Set Theory, Measure Theory, Finite-dimensional Vector Spaces, and a number of other books including his automathography (as Halmos referred to it) I Want to be a Mathematician. Halmos was also a frequent commenter on the craft of disseminating mathematics with articles like “How to write mathematics”, “What to publish” and “How to talk mathematics”.

Turning to the book in question: it is both a mathology, a collection of articles by a variety of mathematicians, and a tribute to Halmos from the Operator Theory community. The first three articles are reminiscences about Halmos that appeared in mathematics journals soon after his death in 2006. More about the man than his mathematics, in these articles some of his closest collaborators and friends impart a feeling for the personality and character of Paul Halmos. Included are a number of quotes from Halmos and excerpts from his writings, which are often both humorous and enlightening.

After a short diversion with eight pages of photographs of Halmos through the years, and some 25 pages of photographs taken by Halmos of other mathematicians (especially ones who figured prominently in his life and/or this mathology) we come to the mathematical meat of the book: fourteen expository articles on some of the key areas of Halmos’s research in Operator Theory. Included are articles on Dilation Theory, Toeplitz Operators, Invariant Subspaces, Subnormal Operators, Hyponormal Operators, Essentially Normal Operators, Commutant Lifting, and the Halmos Similarity Problem. In all of these areas, Halmos was notable at the inception, either with a probing question, illustrative example and/or foundational result. In each article, connections are made from Halmos’s early contributions, to how they led to significant advances by other mathematicians, and to the current research landscape in that area.

If you are interested in Halmos the man and mathematician, I would recommend you first peruse some of his own writings, but if you want to delve deeper into the research topic that was central to his career, this book is an excellent place to start.

Any researcher starting out in Operator Theory, Operator Algebras or a related area would also be well-advised to read this volume. It gives an excellent portrayal the sweep of history in a mathematical discipline and how it can be affected by one man.

Gordon MacDonald is a Professor of Mathematics at the University of Prince Edward Island, Canada, and a “mathematical grandson” of Paul Halmos.