Spectral theory of compact operators
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do not appear in the finite-dimensional case, i.e. for matrices. The compact operators are notable in that they share as much similarity with matrices as one can expect from a general operator. In particular, the spectral properties of compact operators resemble those of square matrices.
This article first summarizes the corresponding results from the matrix case before discussing the spectral properties of compact operators. The reader will see that most statements transfer verbatim from the matrix case.
The spectral theory of compact operators was first developed by F. Riesz.
The classical result for square matrices is the Jordan canonical form, which states the following:
Theorem. Let A be an n × n complex matrix, i.e. A a linear operator acting on Cn. If λ1...λk are the distinct eigenvalues of A, then Cn can be decomposed into the invariant subspaces of A
The subspace Yi = Ker(λi − A)m where Ker(λi − A)m = Ker(λi − A)m+1. Furthermore, the poles of the resolvent function ζ → (ζ − A)−1 coincide with the set of eigenvalues of A.
Let X be a Banach space, C be a compact operator acting on X, and σ(C) be the spectrum of C. The spectral properties of C are:
Theorem.
i) Every nonzero λ ∈ σ(C) is an eigenvalue of C.
ii) For all nonzero λ ∈ σ(C), there exist m such that Ker(λi − C)m = Ker(λi − C)m+1, and this subspace is finite-dimensional.
iii) The eigenvalues can only accumulate at 0. If the dimension of X is not finite, then σ(C) must contain 0.
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