The twofold Ellis-Gohberg inverse problem in an abstract setting and applications

TL;DR

This paper addresses a twofold Ellis-Gohberg inverse problem within an abstract *-algebra framework, providing necessary and sufficient conditions for solutions, proving uniqueness, and exploring special cases including matrix-valued functions on the real line and unit circle.

Contribution

It introduces a novel abstract algebraic approach to the twofold Ellis-Gohberg inverse problem, deriving explicit inversion formulas and extending previous results to new special cases.

Findings

Necessary and sufficient conditions for solution existence

Uniqueness of solutions under natural assumptions

Explicit inversion formula for a block operator matrix

Abstract

In this paper we consider a twofold Ellis-Gohberg type inverse problem in an abstract *-algebraic setting. Under natural assumptions, necessary and sufficient conditions for the existence of a solution are obtained, and it is shown that in case a solution exists, it is unique. The main result relies strongly on an inversion formula for a $2\times 2$ block operator matrix whose off diagonal entries are Hankel operators while the diagonal entries are identity operators. Various special cases are presented, including the cases of matrix-valued $L^1$-functions on the real line and matrix-valued Wiener functions on the unit circle of the complex plane. For the latter case, it is shown how the results obtained in an earlier publication by the authors can be recovered.