Positive and generalized positive real lemma for slice hyperholomorphic functions

TL;DR

This paper establishes a quaternionic positive real lemma and its generalization for slice hyperholomorphic functions, extending classical results to quaternionic analysis with applications to functions with positive real parts and Schur functions.

Contribution

It introduces a quaternionic positive real lemma and a generalized version for slice hyperholomorphic functions, expanding the theoretical framework in quaternionic analysis.

Findings

Proved a quaternionic positive real lemma.

Extended the lemma to cases with negative squares in the kernel.

Applied results to functions with positive real part and Schur functions.

Abstract

In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball.