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

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.