Pick's Theorems for Dissipative Operators

Abstract

Let $\mathcal H$ be a complex Hilbert space and let $A$ be a boundedlinear transformation on $\mathcal H$. For a complex-valuedfunction $f$, which is analytic in a domain $\mathbb{D}$ of the complexplane containing the spectrum of $A$, let $f(A)$ denote theoperator on $\mathcal H$ defined by means of the \textit{Riesz-Dunfordintegral.} In the present paper, several (presumably new) versions of Pick'stheorems are proved for $f(A),$ where $A$ is a dissipativeoperator (or a proper contraction) and $f$ is a suitable analyticfunction in the domain $\mathbb{D}$.