The twofold ellis–gohberg inverse problem in an abstract setting and applications

S. ter Horst; M. A. Kaashoek; F. van Schagen

doi:10.1007/978-3-030-11614-9_8

The twofold ellis–gohberg inverse problem in an abstract setting and applications

S. ter Horst^*, M. A. Kaashoek, F. van Schagen

^*Corresponding author for this work

Mathematics

Research output: Chapter in Book / Report / Conference proceeding › Chapter › Academic › peer-review

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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 × 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 matrixvalued 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.

Original language	English
Title of host publication	Interpolation and Realization Theory with Applications to Control Theory
Subtitle of host publication	In honor of Joe Ball
Editors	V. Bolotnikov, S. ter Horst, A.C.M. Ran, V. Vinnikov
Publisher	Springer International Publishing AG
Pages	155-212
Number of pages	58
ISBN (Electronic)	9783030116149
ISBN (Print)	9783030116132
DOIs	https://doi.org/10.1007/978-3-030-11614-9_8
Publication status	Published - 2019

Publication series

Name	Operator Theory: Advances and Applications
Volume	272
ISSN (Print)	0255-0156
ISSN (Electronic)	2296-4878

Funding

This work is based on the research supported in South Africa (Grant Numbers 90670 and 93406). From the first part of the proof we know that for each i, j the operator Tij maps SB(R+) into SB(R+), and T |SB(R+) is a bounded linear operator from SB(R+) into SB(R+). Now recall that T is given by (A.25). It follows that T maps SB(R+)p into SB(R+)q, and T |SB(R+)p is a bounded linear operator from SB(R+)p into SB(R+)q, which completes the proof. □ Acknowledgement. This work is based on the research supported in part by the National Research Foundation of South Africa. Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard.

Funders	Funder number
National Research Foundation
National Research Foundation

Keywords

Abstract Toeplitz and Hankel operators
Integral operators
Inverse problem
Operator inversion
Wiener functions

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10.1007/978-3-030-11614-9_8

The Twofold Ellis–Gohberg Inverse Problem in an Abstract Setting and ApplicationsFinal published version, 803 KBLicence: Article 25fa Dutch Copyright Act

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ter Horst, S., Kaashoek, M. A., & van Schagen, F. (2019). The twofold ellis–gohberg inverse problem in an abstract setting and applications. In V. Bolotnikov, S. ter Horst, A. C. M. Ran, & V. Vinnikov (Eds.), Interpolation and Realization Theory with Applications to Control Theory: In honor of Joe Ball (pp. 155-212). (Operator Theory: Advances and Applications; Vol. 272). Springer International Publishing AG. https://doi.org/10.1007/978-3-030-11614-9_8

ter Horst, S. ; Kaashoek, M. A. ; van Schagen, F. / The twofold ellis–gohberg inverse problem in an abstract setting and applications. Interpolation and Realization Theory with Applications to Control Theory: In honor of Joe Ball. editor / V. Bolotnikov ; S. ter Horst ; A.C.M. Ran ; V. Vinnikov. Springer International Publishing AG, 2019. pp. 155-212 (Operator Theory: Advances and Applications).

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ter Horst, S, Kaashoek, MA & van Schagen, F 2019, The twofold ellis–gohberg inverse problem in an abstract setting and applications. in V Bolotnikov, S ter Horst, ACM Ran & V Vinnikov (eds), Interpolation and Realization Theory with Applications to Control Theory: In honor of Joe Ball. Operator Theory: Advances and Applications, vol. 272, Springer International Publishing AG, pp. 155-212. https://doi.org/10.1007/978-3-030-11614-9_8

The twofold ellis–gohberg inverse problem in an abstract setting and applications. / ter Horst, S.; Kaashoek, M. A.; van Schagen, F.
Interpolation and Realization Theory with Applications to Control Theory: In honor of Joe Ball. ed. / V. Bolotnikov; S. ter Horst; A.C.M. Ran; V. Vinnikov. Springer International Publishing AG, 2019. p. 155-212 (Operator Theory: Advances and Applications; Vol. 272).

Research output: Chapter in Book / Report / Conference proceeding › Chapter › Academic › peer-review

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N2 - 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 × 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 matrixvalued 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.

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ter Horst S, Kaashoek MA, van Schagen F. The twofold ellis–gohberg inverse problem in an abstract setting and applications. In Bolotnikov V, ter Horst S, Ran ACM, Vinnikov V, editors, Interpolation and Realization Theory with Applications to Control Theory: In honor of Joe Ball. Springer International Publishing AG. 2019. p. 155-212. (Operator Theory: Advances and Applications). Epub 2019 Apr 9. doi: 10.1007/978-3-030-11614-9_8

The twofold ellis–gohberg inverse problem in an abstract setting and applications

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