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A Toeplitz-Like Operator with Rational Matrix Symbol Having Poles on the Unit Circle: Fredholm Properties

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Abstract

This paper concerns the analysis of an unbounded Toeplitz-like operator generated by a rational matrix function having poles on the unit circle \({\mathbb T}\). It extends the analysis of such operators generated by scalar rational functions with poles on \({\mathbb T}\) found in Groenewald et al. (Oper Theory Adv Appl 271:239–268, 2018; Oper Theory Adv Appl 272:133–154, 2019; Integr Equ Oper Theory 91, 2019). A Wiener–Hopf type factorization of rational matrix functions with poles and zeroes on \({\mathbb T}\) is proved and then used to analyze the Fredholm properties of such Toeplitz-like operators. A formula for the index, based on the factorization, is given. Furthermore, it is shown that the determinant of the matrix function having no zeroes on \({\mathbb T}\) is not sufficient for the Toeplitz-like operator to be Fredholm, in contrast to the classical case.

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Acknowledgements

This work is based on research supported in part by the National Research Foundation of South Africa (NRF) 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.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, North-West University, Research Focus Area: Pure and Applied Analytics, Private Bag X6001, Potchefstroom, 2520, South Africa

    G. J. Groenewald & S. ter Horst

  2. DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Potchefstroom, South Africa

    S. ter Horst

  3. Numeracy Center, University of Cape Town, Rondebosch, Cape Town, 7701, South Africa

    J. Jaftha

  4. Department of Mathematics, Faculty of Science, VU Amsterdam, De Boelelaan 1111, 1081 HV, Amsterdam, The Netherlands

    A. C. M. Ran

  5. Research Focus Area: Pure and Applied Analytics, North-West University, Potchefstroom, South Africa

    A. C. M. Ran

Authors
  1. G. J. Groenewald
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  2. S. ter Horst
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  3. J. Jaftha
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  4. A. C. M. Ran
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Corresponding author

Correspondence to S. ter Horst.

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Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Communicated by Seppo Hassi.

Dedicated to our friend an colleague Henk the Snoo, on the occasion of his seventy-fifth birthday.

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This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers 118513 and 127364).

This article is part of the topical collection “Recent Developments in Operator Theory - Contributions in Honor of H.S.V. de Snoo” edited by Jussi Behrndt and Seppo Hassi.

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Groenewald, G.J., ter Horst, S., Jaftha, J. et al. A Toeplitz-Like Operator with Rational Matrix Symbol Having Poles on the Unit Circle: Fredholm Properties. Complex Anal. Oper. Theory 15, 1 (2021). https://doi.org/10.1007/s11785-020-01040-z

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