Schur class operator functions and automorphisms of Hardy algebras

Abstract

Let $E$ be a $W^{\ast }$-correspondence over a von Neumann algebra $M$ and let $H^{\infty }(E)$ be the associated Hardy algebra. If $\sigma$ is a faithful normal representation of $M$ on a Hilbert space $H$, then one may form the dual correspondence $E^{\sigma }$ and represent elements in $H^{\infty }(E)$ as $B(H)$-valued functions on the unit ball $\mathbb{D}(E^{\sigma }{)}^{\ast }$. The functions that one obtains are called Schur class functions and may be characterized in terms of certain Pick-like kernels. We study these functions and relate them to system matrices and transfer functions from systems theory. We use the information gained to describe the automorphism group of $H^{\infty }(E)$ in terms of special Möbius transformations on $\mathbb{D}(E^{\sigma })$. Particular attention is devoted to the $H^{\infty }$-algebras that are associated to graphs.