On linear fractional transformations associated with generalized J-inner matrix functions

TL;DR

This paper investigates generalized J-inner matrix functions, their associated reproducing kernel spaces, and factorization formulas, advancing understanding of their role in indefinite interpolation and Schur class transformations.

Contribution

It introduces the notion of associated pairs for a subclass of generalized J-inner functions and derives new factorization formulas for these functions.

Findings

Reproducing kernel indefinite inner product spaces are characterized.

Factorization formulas for a subclass of generalized J-inner functions are established.

The role of these functions in indefinite interpolation problems is clarified.

Abstract

In this paper we study generalized J-inner matrix valued functions which appear as resolvent matrices in various indefinite interpolation problems. Reproducing kernel indefinite inner product spaces associated with a generalized J-inner matrix valued function W are studied and intensively used in the description of the range of the linear fractional transformation associated with W and applied to the Schur class. For a subclass of generalized J-inner matrix valued function W the notion of associated pair is introduced and factorization formulas for W are found.