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A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm characteristics.
- Source :
-
Linear Algebra & its Applications . Sep2024, Vol. 697, p155-183. 29p. - Publication Year :
- 2024
-
Abstract
- In a recent paper (Groenewald et al. 2021 [9]) we considered an unbounded Toeplitz-like operator T Ω generated by a rational matrix function Ω that has poles on the unit circle T of the complex plane. A Wiener-Hopf type factorization was proved and this factorization was used to determine some Fredholm properties of the operator T Ω , including the Fredholm index. Due to the lower triangular structure (rather than diagonal) of the middle term in the Wiener-Hopf type factorization and the lack of uniqueness, it is not straightforward to determine the dimension of the kernel of T Ω from this factorization, and hence of the co-kernel, even when T Ω is Fredholm. In the current paper we provide a formula for the dimension of the kernel of T Ω under an additional assumption on the Wiener-Hopf type factorization. In the case that Ω is a 2 × 2 matrix function, a characterization of the kernel of the middle factor of the Wiener-Hopf type factorization is given and in many cases a formula for the dimension of the kernel is obtained. The characterization of the kernel of the middle factor for the 2 × 2 case is partially extended to the case of matrix functions of arbitrary size. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 697
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 177877521
- Full Text :
- https://doi.org/10.1016/j.laa.2023.09.028