Abstract
This paper deals with the inverse problem for the class of orthogonal functions that for the scalar case was introduced by Ellis and Gohberg (J Funct Anal 109:155–198, 1992). The problem is reduced to a linear equation with a special right hand side. This reduction allows one to solve the inverse problem for square matrix functions under conditions that are natural generalizations of those appearing in the scalar case. These conditions lead to a unique solution. Special attention is paid to the polynomial case. A number of partial results are obtained for the non-square case. Various examples are given to illustrate the main results and some open problems are presented.
Similar content being viewed by others
References
Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators. Akademie-Verlag/Springer-Verlag, Berlin/Berlin (1990)
Ellis, R. L.: An Identity Satisfied by Certain Orthogonal Vector-Valued Functions. In: A Panorama of Modern Operator Theory and Related Topics. The Israel Gohberg Memorial Volume. Oper. Theory Adv. Appl., vol.218. Birkhäuser Verlag, Basel, pp. 329–344 (2012)
Ellis R.L., Gohberg I.: Orthogonal systems related to infinite Hankel matrices. J. Funct. Anal. 109, 155–198 (1992)
Ellis R.L., Gohberg I.: Orthogonal Systems and Convolution Operators. Oper. Theory Adv. Appl., vol. 140. Birkhäuser Verlag, Basel (2003)
Ellis R.L., Gohberg I., Lay D.C.: Infinite analogues of block Toeplitz matrices and related orthogonal functions. Integr. Equ. Oper. Theory 22, 375–419 (1995)
Ellis R.L., Gohberg I., Lay D.C.: On a class of block Toeplitz matrices. Linear Algebra Appl. 241–243, 225–245 (1996)
Frazho A.E., Bosri W.: An operator Perspective on Signals and Systems. Oper. Theory Adv. Appl., vol. 204. Birkhäuser Verlag, Basel (2010)
Gohberg I., Goldberg S., Kaashoek M.A.: Classes of Linear Operators, vol. II. Oper. Theory Adv. Appl., vol. 63. Birkhäuser Verlag, Basel (1993)
Gohberg I., Kaashoek M.A., van Schagen F.: Partially Specified Matrices and Operators: Classification, Completion, Applications. Oper. Theory Adv. Appl., vol. 79. Birkhäuser Verlag, Basel (1995)
Kaashoek M.A., van Schagen F.: Ellis–Gohberg identities for certain orthogonal functions I: block matrix generalizations and ℓ 2-setting. Indag. Math. 23, 777–795 (2012)
Kaashoek M.A., van Schagen F.: Ellis-Gohberg identities for certain orthogonal functions II: Algebraic setting and asymmetric versions. West Memorial Issue. Math. Proc. R. Irish Acad. 113A, 107–129 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kaashoek, M.A., van Schagen, F. The Inverse Problem for Ellis–Gohberg Orthogonal Matrix Functions. Integr. Equ. Oper. Theory 80, 527–555 (2014). https://doi.org/10.1007/s00020-014-2159-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-014-2159-4