Generalized quaternionic Schur functions in the ball and half-space and Krein-Langer factorization

TL;DR

This paper extends the Krein-Langer factorization theorem to the setting of slice hyperholomorphic quaternionic functions, covering functions with negative squares on quaternionic balls and half-spaces, using advanced functional analysis tools.

Contribution

It provides a more general version of the Krein-Langer factorization theorem in the quaternionic slice hyperholomorphic setting, applicable to broader classes of functions.

Findings

Established a new Krein-Langer factorization in the quaternionic setting.

Extended the theory to functions with negative squares on quaternionic domains.

Utilized Schauder-Tychonoff and invariant subspace theorems in the proof.

Abstract

In this paper we prove a new version of Krein-Langer factorization theorem in the slice hyperholomorphic setting which is more general than the one proved in [D. Alpay, F. Colombo, I. Sabadini, Krein-Langer factorization and related topics in the slice hyperholomorphic setting, J. Geom. Anal., 24 (2014), 843--872]. We treat both the case of functions with $\kappa$ negative squares defined on subsets of the quaternionic unit ball or on subsets of the half space of quaternions with positive real part. A crucial tool in the proof of our results is the Schauder-Tychonoff theorem and an invariant subspace theorem for contractions in a Pontryagin space.