In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets.
In general, operators on infinite-dimensional spaces feature properties that do not appear in the finite-dimensional case, i.e. for matrices.
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.
Furthermore, the poles of the resolvent function ζ → (ζ − A)−1 coincide with the set of eigenvalues of A. Theorem — Let X be a Banach space, C be a compact operator acting on X, and σ(C) be the spectrum of C. The theorem claims several properties of the operator λ − C where λ ≠ 0.
This fact will be used repeatedly in the argument leading to the theorem.
Notice that when X is a Hilbert space, the lemma is trivial.
The same argument goes through if the distances d(xn, Ker(I − C)) is bounded.
, and {xn} are now viewed as representatives of their equivalence classes in the quotient space.
> k and define a sequence of unit vectors by znk = xnk
This is impossible because z is the norm limit of a sequence of unit vectors.
By Lemma 2, Y1 = Ran(I − C) is a closed proper subspace of X.
Define Yn = Ran(I − C)n. Consider the decreasing sequence of subspaces where all inclusions are proper.
As before, compactness of C means {C yn} must contain a norm convergent subsequence.
iii) Suppose there exist infinite (at least countable) distinct eigenvalues {λn}, with corresponding eigenvectors {xn}, such that
So we have that there are only finite distinct eigenvalues outside any ball centered at zero.
The set of eigenvalues {λ} is the union Because σ(C) is a bounded set and the eigenvalues can only accumulate at 0, each Sn is finite, which gives the desired result.
v) As in the matrix case, this is a direct application of the holomorphic functional calculus.
As in the matrix case, the above spectral properties lead to a decomposition of X into invariant subspaces of a compact operator C. Let λ ≠ 0 be an eigenvalue of C; so λ is an isolated point of σ(C).
Using the holomorphic functional calculus, define the Riesz projection E(λ) by where γ is a Jordan contour that encloses only λ from σ(C).
C restricted to Y is a compact invertible operator with spectrum {λ}, therefore Y is finite-dimensional.
If B is an operator on a Banach space X such that Bn is compact for some n, then the theorem proven above also holds for B.