Spectral Decomposition of Compact Operators on Hilbert Spaces

Abstract

Compact operators are linear operators on Banach spaces that maps bounded set to relatively compact sets. In the case of Hilbert space H it is an extension of the concept of matrix acting on a finite dimensional vector space. In Hilbert space, compact operators are the closure of the finite rank operators in the topology induced by the operator norm. In general, operators on infinite dimensional spaces feature properties that do not appear in the finite dimension case; i.e matrices. The compact operators are notable in that they share as much similarity with matrices as one can expect from a general operator. Spectral decomposition of compact operators on Banach spaces takes the form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, the spectral properties of compact operators resembles those of square matrices.