Pick’s theorems for dissipative operators

Abstract

Let H be a complex Hilbert space and let A be a bounded linear transformation on H. For a complex-valued function f, which is analytic in a domain D of the complex plane containing the spectrum of A, let f (A) denote the operator on H defined by means of the Riesz-Dunford integral. In the present paper, several (presumably new) versions of Pick?s theorems are proved for f (A), where A is a dissipative operator (or a proper contraction) and f is a suitable analytic function in the domain D.