Schur functions and their realizations in the slice hyperholomorphic setting

TL;DR

This paper extends Schur analysis to the quaternionic setting using slice hyperholomorphic functions, introducing realizations via the S-resolvent operator and exploring reproducing kernels, positive definite functions, and Schur multipliers.

Contribution

It develops a quaternionic Schur analysis framework using slice hyperholomorphic functions, extending complex results with new realizations and operator models.

Findings

Realizations expressed through the S-resolvent operator.

Extension of reproducing kernel and positive definite function theory.

Characterization of Schur multipliers and their co-isometric realizations.

Abstract

we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels, positive definite functions in this setting and we show how they can be obtained in our setting using the extension operator and the slice regular product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges-Rovnyak space.