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.