Pontryagin de Branges Rovnyak spaces of slice hyperholomorphic functions

TL;DR

This paper explores the theory of slice hyperholomorphic functions within reproducing kernel Hilbert and Pontryagin spaces, introducing quaternionic analogs of classical analytic function spaces, and addressing interpolation and realization problems.

Contribution

It develops the theory of Pontryagin and Hilbert spaces of slice hyperholomorphic functions, including Hardy spaces, Blaschke products, and realization theorems in the quaternionic setting.

Findings

Established quaternionic Hardy space and Blaschke products.

Proved key results for Pontryagin spaces in the quaternionic context.

Developed realization theory for Schur and Carathéodory functions.

Abstract

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, a theorem of Shmulyan on densely defined contractive linear relations. We then study realizations of generalized Schur functions and of generalized Carath'eodory functions.