Multivariable generalizations of the Schur class: positive kernel characterization and transfer function realization

TL;DR

This paper extends the classical Schur class to multivariable operator-valued functions using reproducing kernel Hilbert correspondences, providing new realization theorems and deepening the theoretical understanding of these generalized classes.

Contribution

It introduces the concept of reproducing kernel Hilbert correspondences and links Muhly-Solel Hardy spaces to these kernels, enriching the theory of multivariable Schur classes.

Findings

Reproducing kernel Hilbert correspondences are used to characterize multivariable Schur classes.

The paper establishes new realization theorems for these classes in special cases.

It deepens the analogy between classical and multivariable Schur classes.

Abstract

The operator-valued Schur-class is defined to be the set of holomorphic functions $S$ mapping the unit disk into the space of contraction operators between two Hilbert spaces. There are a number of alternate characterizations: the operator of multiplication by $S$ defines a contraction operator between two Hardy Hilbert spaces, $S$ satisfies a von Neumann inequality, a certain operator-valued kernel associated with $S$ is positive-definite, and $S$ can be realized as the transfer function of a dissipative (or even conservative) discrete-time linear input/state/output linear system. Various multivariable generalizations of this class have appeared recently,one of the most encompassing being that of Muhly and Solel where the unit disk is replaced by the strict unit ball of the elements of a dual correspondence $E^{\sigma}$ associated with a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $\cA$ together with a $*$-representation $\sigma$ of $\cA$. The main new point which we add here is the introduction of the notion of reproducing kernel Hilbert correspondence and identification of the Muhly-Solel Hardy spaces as reproducing kernel Hilbert correspondences associated with a completely positive analogue of the classical Szeg\"o kernel. In this way we are able to make the analogy between the Muhly-Solel Schur class and the classical Schur class more complete. We also illustrate the theory by specializing it to some well-studied special cases; in some instances there result new kinds of realization theorems.