On a polyanalytic a approach to noncommutative de Branges-Rovnyak spaces and Schur analysis

TL;DR

This paper introduces a polyanalytic approach to noncommutative de Branges-Rovnyak spaces and Schur analysis within Fueter hyperholomorphic functions, connecting operator theory with slice polyanalytic functions.

Contribution

It develops a novel framework using Appell-like polynomials for Schur analysis and de Branges-Rovnyak spaces in hyperholomorphic function theory, expanding existing methods.

Findings

Describes a Hardy space in the hyperholomorphic setting

Characterizes Schur multipliers and related kernels

Analyzes Blaschke and Herglotz functions in this framework

Abstract

In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carath\'eodory multipliers.