Spectral Theory of Compact Self Adjoint Operators

Abstract

Let H be a separable Hilbert space, and let B(H) and C(H) denote the bounded linear operators on H and the compact operators on H, respec-tively. 1 The Resolvent Set and the Spectrum of an Operator For n×n matrices, the spectrum is just the set of eigenvalues. The spectrum of a linear operator L is defined indirectly, as the complement of another set, the resolvent set. It is necessary to do this because on an infinite dimensional space the operator L may not have eigenvalues in the usual sense. Definition 1.1. Let L ∈ B(H). The resolvent set of L is ρ(L) := {λ ∈ C : (L − λI) −1 ∈ B(H)} 1 . The operator R L (λ) := (L − λI) −1 is called the resolvent of L. The spectrum of L, σ(L), is defined as the complement of the resolvent set: σ(L) := ρ(L) . This agrees with the definition of the spectrum in the matrix case, where the resolvent set comprises all complex numbers that are not eigenvalues. In terms of its spectrum, we will see that a compact operator behaves like a matrix, in the sense that its spectrum is the union of all of its eigenvalues and 0. We begin with the eigenspaces of a compact operator. We start with two lemmas that we will need in the sequel. The first holds for all self-adjoint operators, including unbounded ones. Lemma 1.2. Let L = L * be in B(H). Then the eigenvalues of L are real and the eigenvectors corresponding to distinct eigenvalues are orthogonal.