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Spectral Theory of Operators on Hilbert Spaces

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A colleague of mine claims that the spectral theorem is the most beautiful theorem in mathematics. To be sure, such a game of favorites is always a dicey one to play: it pretty much comes down to such things as deciding which of Beethoven’s symphonies, or which of Michelangelo’s statues, is the best. But it’s certainly the case that just about every one you might ask will come up with the same top three in each category: Beethoven’s 5th, 9th, 7th tend to race to the front of the pack, as do Michelangelo’s Pietà, David, and Moses. Likewise it’s possible (and possibly somewhat less dicey) to list, say, six (this being a perfect number) theorems that are along the lines of universal favorites, leaving out the older classics such as the Pythagorean Theorem, the irrationality of √2, or even Cantor’s proofs regarding infinite sets. At the risk of causing trouble I suggest the favored sextet should be as follows, in no particular order: the first isomorphism theorem of Emmy Noether, the de Rham - Hodge theorem, the theorem of quadratic reciprocity (of Gauss and Euler), the Lebesgue dominated convergence theorem, the proof of the irrationality of π, and yes, the spectral theorem. It’s all truly beautiful stuff.

I acknowledge that I am letting myself in for an avalanche of shouts of “Yes, but what about …” — to wit: “Hey, you goof, what about Fermat’s Last Theorem, then?” or “Where the hell is Bézout?” or “What’s wrong with Gauss-Bonnet? — the list goes on (and the mind boggles). So let me just say that I’m restricting myself to stuff we can safely show the kids in a senior level undergraduate course for extremely enthusiastic majors. When I was a snot, we got the vaunted elementary proof of the Prime Number Theorem à la Erdös-Selberg (given that my professor was the late Ernst Straus, a close friend of “Uncle Paul” himself), but that took nigh on 2/3 of the term: too involved a proof, I think. On the other hand, there’s the gorgeous proof by Newman… but I am getting way off the subject: we’re making a case for the spectral theorem — for which no one really needs to make a case, anyway — so there!

Well, at its most prosaic level, and as every one knows, the spectral theorem gives us the diagonalization of matrices in the presence of the eigenvalue problem, but at a much more powerful level it is this theorem which resides at the heart of, for example, the now-classical treatment of the Schrödinger wave equation in quantum mechanics. Indeed, along the latter lines the spectral theorem also perfectly fits the bill regarding Dirac’s famous pronouncement that the mathematics of physics must be beautiful because, after all, it was chosen by God: it is truly awe-inspiring to learn that the quantized data provided by the Schrödinger equation’s solution via spectral theory is precisely what the experiments provide, suggesting that the overarching mathematics for quantum mechanics should be nothing else than functional analysis in Hilbert space (as developed largely by John von Neumann, of course). As I never tire of telling my students: it’s better than magic — it’s mathematics.

With this observation in place, what about the author’s own appraisal of what he is presenting in the book under review? Well, in the introductory paragraph to his (thirty-some pages long) third chapter titled “Spectral Theorem” Kubrusly states that

[t]he Spectral Theorem is one of the milestones in the theory of Hilbert space operators, providing a full statement about the structure of normal operators. For compact operators the Spectral Theorem can be completely investigated without requiring any knowledge of measure theory, and this leads to the concept of diagonalization. However, the Spectral Theorem for plain normal operators (the general case) requires some (elementary) measure theory.

There’s a lot going on in the shadows, of course. Properly speaking the spectral theorem asserts that a normal operator T on a Hilbert space allows for a positive measure on an associated sigma algebra of subsets of a suitable set Ω, together with a function φ depending on T of course, such that T is unitarily equivalent to the operator “multiplication by φ” on the space L2(Ω), and T and φ have the same norm (see p.67 of the book). Actually Kobrusly calls this the “first version” of the theorem; see p.76 for his “second version” which associates T, as above, to a unique spectral measure such that T can be realized in terms of integration against this measure. Standard as these theorems are, this is deep stuff under any circumstances and clearly requires a measure of sophistication on the part of the reader (if you’ll excuse the pun).

But it’s all there in Kobrusly’s book, done very carefully and thoroughly, and he situates the spectral theorem in a very interesting broader context. For instance, he gives us Fuglede’s Theorem which he presents in the language of spectral measures, and he does some C*-algebra and some Riesz functional calculus, while adding an entire (closing) chapter on Fredholm theory.

Pedagogically speaking, Kubrusly intends the book for a one-semester graduate course and even though exercises are absent, he closes each of his five (pretty beefy) chapters with sections titled “Additional Propositions.” Says he:

[These consist] of either auxiliary results that will be required to support a proof in the main text, or further results extending some theorems proved in the main text. These are followed by a set of Notes, where each proposition is briefly discussed, and references are provided indicating proofs for all of them. The Additional Propositions can … be thought of as a set of proposed problems, and their … Notes … as hints for solving them.

I think this is a nice idea: it will make for a very solid learning experience indeed. All in all, it’s a good book.

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

Date Received: 
Thursday, July 26, 2012
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Carlos S. Kubrusly
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Michael Berg
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Monday, August 20, 2012