What is the spectral theorem for compact self-adjoint operators on a Hilbert space actually for?

Question

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.)

Answer 1

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.)

Answer 2

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.

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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 :-)

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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...)

Answer 3

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.