The Elements of Operator Theory

There is something particularly appealing about a mathematics text that starts off with a quote by Graham Greene. It is a bit of an enigmatic one, I admit: “The truth … has never been of any real value to any human being — It is a symbol for mathematicians and philosophers to pursue. In human relations kindness and lies are worth a thousand truths …” Unless the author of the book under review, Carlos Kubrusly, of Rio de Janeiro’s Catholic University, is cynical to the point of making Kafka look like an optimist, the quote is surely meant to be ironic, so let’s just stipulate this as being the case. But it is unquestionably also the case that for us mathematicians the truth of things has far, far greater resonance than it might for most every one else — it’s truly a virtuous obsession.

But philosophical reflections and soul-searching are not the order of this day: what about Kubrusly’s book? Well, it’s quite good. While Kubrusly is attached to PUC–Rio’s department of electrical engineering, he has evidently kept his mathematical spirit pure, seeing that the book shown no signs whatsoever of the evils engineers tend to visit on mathematics: after all, in their relations expediency and vagueness are (all too often) worth a thousand rigors (pace Graham Greene). Fear not, Kubrusly’s immunity is apparently total. (I guess I am betraying something of a personal subtext in all this: I have been part of a physics working group recently and have begun to notice certain alarming tendencies in myself … Time to take drastic measures!)

The Elements of Operator Theory is a very good treatment of some of the mainstays of functional analysis. It’s the book’s second edition, the first dating back a decade. But it is also quite an unusual book in the sense that it starts with very elementary material indeed, and, proceeding linearly (says Kubrusly: “The logical dependence of the various sections (and chapters) … reflects approximately the minimum amount of material to proceed further …”), takes the reader all the way to the spectral theorem, or at least through the case of compact operators (and compact normal operators). Kubrusly finishes the book with “A Glimpse at the General Case.” This is consistent with the fact that while he explicitly aims his treatment at graduate students in mathematics, Kubrusly is also concerned with providing a self-contained reference to fellow travelers; see below for his remarks about the book’s genesis.

In an orbit of some 500 pages, then, starting out with a chapter or two on set theoretic and linear algebraic basics (Kubrusly’s second chapter is essentially a crash course in vector spaces pitched a little, but not much, below the level of difficulty of Halmos’ Finite Dimensional Vector Spaces), the next order of business is — properly, of course — topology, and obviously the metric space material relevant to main stream functional analysis enjoys a certain primacy here. The level of difficulty can be gauged by the fact that this third chapter closes with a discussion of compactness vis à vis sequential compactness.

The book’s meat and potatoes is located in the latter trio of chapters, 300 pages’ worth: Banach spaces, Hilbert spaces, and the Spectral Theorem. We encounter “all the usual suspects” in Kubrusly’s discussion: open mapping, Banach-Steinhaus, Hahn-Banach, unitarity, Riesz representation, adjointness, and then spectral theory.

A minor caveat is in order, however. As far as how The Elements of Operator Theory might be used at American Universities is concerned, a little care must be taken. Consider, e.g., the following fragment from p. 193: “The failure of R^p[0,1] to be complete when equipped with its usual metric d_p is regarded as one of the defects of the Riemann integral. A more general concept of integral, viz., the Lebesgue integral, corrects this and other drawbacks of the Riemann integral.” However, the Lebesgue integral’s properties (and R^p[0,1]’s denseness in L^p[0,1]) are only sketched; conclusion: Kubrusly’s treatment is a bit more elementary than that in the vast majority of texts on this material, say, Halmos’ books, or certainly Rudin’s Functional Analysis.

However, as I already suggested above, Kubrusly’s treatment is wonderful. The book is very thorough and clear, and very accessible: it is easy to read even as the material discussed is treated very seriously. Indeed, it stands to reason that this should be so: “I [Kubrusly] started writing this book after lecturing on its subject at Catholic University of Rio de Janeiro for over 20 years. In general the material is covered in two one-semester beginning graduate courses, where the audience comprises mathematics, engineering, economics, and physics students. Quite often senior undergraduate students joined the courses …” This ecumenism is of course consistent with the fact that Kubrusly’s first two chapters are of so elementary a nature.

All of this translates to how the book might be used at American Universities. It certainly would serve beautifully as a text for advanced undergraduates. For beginning graduate students, its utility certainly resides in its clarity and accessibility: I would recommend additional work on the Lebesgue integral sooner rather than later (is Burkill’s little book still available?). Kubrusly includes on the order of 300 exercises in the book, there are many good examples, and he takes the trouble to add both the right kind and amount of motivating commentary and discussions of what might be hiding around the next corner.

The Elements of Operator Theory is quite a good book.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.