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

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

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 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 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.

Original languageEnglish
Article number1
Pages (from-to)1-29
Number of pages29
JournalComplex Analysis and Operator Theory
Volume15
Issue number1
Early online date13 Nov 2020
DOIs
Publication statusE-pub ahead of print - 13 Nov 2020

Keywords

  • Fredholm properties
  • Rational matrix functions
  • Toeplitz operators
  • Unbounded operators
  • Wiener–Hopf factorization

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