Analytic Functions of Operators

Abstract

We have observed in Section 1.7 that there is a very rich functional calculus for normal operators. We now return to the study of arbitrary bounded operators on Hilbert space and investigate the possibility of defining functions of such an operator. Let A be a bounded operator on ℋ. We already have a definition of p(A)for any complex polynomial p. In this chapter we shall show how to define f (A) whenever f is a function analytic on an open set containing σ(A). The functional calculus which we shall develop is not as rich as the functional calculus for normal operators; nonetheless it has important applications to the study of invariant subspaces. It also has many other important applications which we shall not discuss; (e.g., see Proposition 2.3).