Realizations of holomorphic and slice hyperholomorphic functions: the Krein space case

TL;DR

This paper develops realization theorems for operator-valued functions in both complex and quaternionic settings, utilizing Krein spaces, and introduces the first such theorem with a quaternionic Krein space as the state space.

Contribution

It presents the first realization theorem for quaternionic slice hyperholomorphic functions with a Krein space as the state space, extending complex results to the quaternionic setting.

Findings

Proves a realization theorem for operator-valued functions analytic near the origin in the complex case.

Extends the realization theorem to quaternionic slice hyperholomorphic functions.

Utilizes reproducing kernel Krein spaces to achieve these results.

Abstract

In this work we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space