A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Invertibility and Riccati equations

G. J. Groenewald, S. ter Horst*, J. Jaftha, A. C.M. Ran

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

This paper is a continuation of the work on unbounded Toeplitz-like operators TΩ with rational matrix symbol Ω initiated in Groenewald et al. (2021) [11], where a Wiener-Hopf type factorization of Ω is obtained and used to determine when TΩ is Fredholm and to compute the Fredholm index in case TΩ is Fredholm. Due to the high level of non-uniqueness and complicated form of the Wiener-Hopf type factorization, it does not appear useful in determining when TΩ is invertible. In the present paper we use state space methods to characterize invertibility of TΩ in terms of the existence of a stabilizing solution of an associated nonsymmetric discrete algebraic Riccati equation, which in turn leads to a pseudo-canonical factorization of Ω and concrete formulas of TΩ−1.

Original languageEnglish
Article number127925
Pages (from-to)1-15
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Volume532
Issue number2
Early online date13 Nov 2023
DOIs
Publication statusPublished - 15 Apr 2024

Bibliographical note

Funding Information:
This work is based on research supported in part by the National Research Foundation of South Africa (NRF, Grant Numbers 118513 , 127364 and 145688 ) and the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF and CoE-MaSS do not accept any liability in this regard.

Publisher Copyright:
© 2023 The Author(s)

Funding

This work is based on research supported in part by the National Research Foundation of South Africa (NRF, Grant Numbers 118513 , 127364 and 145688 ) and the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF and CoE-MaSS do not accept any liability in this regard.

Keywords

  • Invertibility
  • Pseudo-canonical factorization
  • Riccati equations
  • Toeplitz operators
  • Unbounded operators

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