Spectral Decomposition of Compact Selfadjoint Operators. Analytic Functions of Operators

Abstract

The well-known theorem from linear algebra on the reducibility of an Hermitian matrix to the diagonal form can be reformulated as follows: For any selfadjoint operator A acting on the finite-dimensional Hilbert space, ℂ N there exists an orthonormal basis (φ k ) k=1 N composed of the eigenvectors of A. The generalization of this result to the case of an operator A = A*, acting on an infinite-dimensional Hilbert space H, by “passing to the limit as N → ∞” is, generally speaking, impossible because a selfadjoint operator may have no eigenvectors in the case where dim H = ∞ (see Example 8.8.1). However, if A is a compact selfadjoint operator, the spectral theorem for A = A* ∈ ℒ(ℂ N ) admits a direct generalization of the indicated type. Section 1 of this chapter deals with the proof of this assertion. In Section 2, we study decompositions in eigenfunctions of selfadjoint integral operators.