Schatten class and Berezin transform of quaternionic linear operators

TL;DR

This paper extends the concepts of Schatten class and Berezin transform to quaternionic operators, providing foundational definitions and properties within weighted Bergman spaces of slice hyperholomorphic functions, based on quaternionic spectral theory.

Contribution

It introduces the Schatten class and Berezin transform for quaternionic operators, linking them to quaternionic spectral theory and weighted Bergman spaces.

Findings

Defined Schatten class for quaternionic operators

Established properties of the Berezin transform in this setting

Connected spectral theory with quaternionic operator analysis

Abstract

In this paper we introduce the Schatten class of operators and the Berezin transform of operators in the quaternionic setting. The first topic is of great importance in operator theory but it is also necessary to study the second one because we need the notion of trace class operators, which is a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S-spectrum for quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S-functional calculus.