Analytic operator functions

Abstract

Let H denote a complex Hilbert space. Let L(H) be the complex Banach space of all bounded linear operators on H, and L(H)*, the conjugate space of L(H). By an operator function f on a domain D, i.e., an opn connected subset of the complex plane, we mean that f(z) E L(H) for every z E D. An operator function f on a domain D is said to be analytic if &f(z)) is analytic on D in the classical sense for every a, E L(H)*. A,(D) will indicate the set of all analytic operator functions on D into L(H). It is well-known that most of the results of complex function theory, such as Cauchy’s integral theorem, Taylor expansion, Cauchy estemates, the maximum modulus principle and so on, are applicable to A,(D). We will appeal to them frequently. For references, we refer the reader to (71. In [5], K. Fan proves the following