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 language | English |
|---|---|
| Pages (from-to) | 1-29 |
| Number of pages | 29 |
| Journal | Complex Analysis and Operator Theory |
| Volume | 15 |
| Issue number | 1 |
| Early online date | 13 Nov 2020 |
| DOIs | |
| Publication status | Published - Feb 2021 |
Funding
This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers 118513 and 127364). 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.
| Funders | Funder number |
|---|---|
| DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences | |
| National Research Foundation | 118513, 127364 |
Keywords
- Fredholm properties
- Rational matrix functions
- Toeplitz operators
- Unbounded operators
- Wiener–Hopf factorization