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arXiv · 2005.14561

A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm properties

G. J. Groenewald·S. ter Horst·J. Jaftha·A. C. M. Ran
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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 T. It extends the analysis of such operators generated by scalar rational functions with poles on T found in [11,12,13]. A Wiener-Hopf type factorization of rational matrix functions with poles and zeroes on 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 T is not sufficient for the Toeplitz-like operator to be Fredholm, in contrast to the classical case.

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G. J. Groenewald, S. ter Horst, J. Jaftha, A. C. M. Ran. 2020-05-29. A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm properties. https://arxiv.org/abs/2005.14561

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