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Please excuse the naive question. I have had two classes now in which this theorem was taught and proven, but I have only ever seen a single (indirect?) application involving the quantum harmonic oscillator. Even if this is not the strongest spectral theorem, it still seems useful enough that there should be many nice examples illustrating its utility. So... what are some of those examples?

(I couldn't readily find any nice examples looking through a few functional analysis textbooks, either. Maybe I have the wrong books.)

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A lot of Hilbert-Schmidt operators are self-adjoint and compact. So, also Fredholm integral equations. These are quite useful in applications. –  Jonas Teuwen Nov 30 '10 at 23:59
Thanks, Jonas. Do you have a good reference with some explicit examples? –  Qiaochu Yuan Dec 1 '10 at 0:01
I know some books that treat integral equations, but do not explicitly mention the spectral theorem (that is usually something for another course...), like Riesz' Functional Analysis (also a nice book in other aspects). It is in a Dover paperback. Further you have Hochstadt's integral equations, from the Wiley Classics. When I'm at the university tomorrow I'll see if there are some books in the analysis library that treat both explicitly. –  Jonas Teuwen Dec 1 '10 at 0:11
Kato's Perturbation Theory for Linear Operators gives nice basic examples that relate operator theory to boundary value problems for differential equations. For the spectral theorem, applications to PDEs are common in mechanics and fluid dynamics, for example. Often the map from data to solution of a boundary-value problem on a bounded domain is compact. –  Bob Pego Dec 1 '10 at 3:23
in sturm-louville theory they say the differential operators from the differential equation taken together forms a new operator that is self adjoint and compact - so it follows from the spectral theorem that there is a basis of orthogonal eigenfunctions that you can compose the solution to the differential equation from! –  Peter Sheldrick Jul 22 '11 at 15:47
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What is the spectral theorem for compact operators good for? Here are some examples. (I am ignoring the self-adjoint aspects, since they don't really play a role in the theorem. And it is valid for more general spaces than Hilbert spaces too, so I will also ignore that part, in the sense that I won't pay too much attention to whether my examples deal with Hilbert spaces on the nose, rather than some variant.)

  • Proving the Peter--Weyl theorem.

  • Proving the Hodge decomposition for cohomology of manifolds (using the fact that the inverse to the Laplacian is compact); Willie noted this example in his answer too.

  • Proving the finiteness of cohomology of coherent sheaves on compact complex analytic manifolds.

  • In its $p$-adic version, the theory of compact operators is basic to the theory of $p$-adic automorphic forms: e.g. in the construction of so-called eigenvarieties parameterizing $p$-adic families of automorphic Hecke eigenforms of finite slope.

  • It is also a basic tool in more classical problems, such as the theory of integral equations. (It is in this context that the theory was first developed; see Dieudonne's book on the history of functional analysis for a very nice account of the historical development of the theory.)

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Can the spectral theorem be applied to show more generally that the higher direct images of a coherent analytic sheaf under a proper map of analytic spaces is coherent? By the way, I'd be very curious if you had a reference for that. –  Akhil Mathew Dec 1 '10 at 4:23
@Akhil: Dear Akhil, That's a good question, to which unfortunately I don't have a good answer. My instinct is to approach it as follows: the question is local on the base, so restricting to a well-chosen n.h. of any given point, we can assume the base is Stein. The point of this is that on a Stein manifold, coherent sheaves are determined by their modules of sections (thought of as Frechet modules over the Frechet algebra of global holomorphic functions). So one could try to resolve the coherent sheaf upstairs in some reasonable way, and then compute the derived pushforward using this ... –  Matt E Dec 1 '10 at 4:38
... resolution, and try to show that the cohomologies are coherent sheaves by studying their global sections as modules over the functions on the base, and get some control over them (here one would use properness, and some relative version of Montel's theorem; and --- if it worked --- one wold be applying a relative version of compact operators, for Frechet modules over the Frechet algebra of functions on the base). I don't know if this actually works, though. –  Matt E Dec 1 '10 at 4:40
I didn't know that it was first used in the theory of integral equations, that is where I use it. Nice answer. –  Jonas Teuwen Dec 1 '10 at 8:59
Thanks for the very informative answer. I've been meaning to check out Dieudonne's history for awhile now and I guess now is a good time. –  Qiaochu Yuan Dec 1 '10 at 10:12
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Maybe a definition of functional calculus? Using quantum mechanics notation, if $A$ is self-adjoint and compact, then $A = \sum \lambda_k |k\rangle\langle k|$, which means that for $f:\mathbb{R}\to\mathbb{R}$ we can define $f(A) = \sum f(\lambda_k) |k\rangle\langle k|$.

This allows us to give a simple demonstration of Stone's theorem for such operators: that $\exp itA$ is a strongly continuous one-parameter unitary group on your Hilbert space.

It also gives very simple motivation for the construction of Green's functions and resolvent operators.

Besides the usual quantum harmonic oscillator, a similar construction can be used to give the decomposition of $L^2$ via eigenfunctions of the Laplacian on a compact manifold. This naturally leads to the Hodge decomposition, which I'm told is generally considered to be somewhat useful :-)

That on a compact manifold, the inverse of the Laplacian is a compact operator means that, discarding the harmonic functions, the Laplacian has a lowest eigenvalue. This fact (and that self-adjointness allows it to be diagonalized) allows you to define the Zeta-function determinant of the Laplacian using some analytic continuation tricks. On 2-dimensional closed surfaces, this is an interesting invariant with nice geometric properties.

(Note that in the case of non-compact domains, the inverse Laplacian is no longer a compact operator, and the Laplacian has a continuous spectrum. So the summation in the zeta-function determinant no longer makes any sense...)

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The only way I know to prove "discreteness" of some piece of a spectrum is to find one or more compact operators on it, suitably separating points. That is, somehow the only tractable operators are those closely related to compact ones.

Even to discuss the spectral theory of self-adjoint differential operators $T$, the happiest cases are where $T$ has compact resolvent $(T-\lambda)^{-1}$.

In particular instances, the Schwartz kernel theorem depends on the compactness of the inclusions of Sobolev spaces into each other (Rellich's lemma).

In automorphic forms: to prove the discreteness of spaces of cuspforms, one shows that the natural integral operators (after Selberg, Gelfand, Langlands et alia) restricted to the space of $L^2$ cuspforms are compact.

One of Selberg's arguments, Bernstein's sketch, Colin de Verdiere's proof, and (apparently) the proof in Moeglin-Waldspurger's book (credited to Jacquet, credited to Colin de Verdiere!?) of meromorphic continuation of Eisenstein series of various sorts depends ultimately on proving compactness of an operator.

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