## 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.

Original language | English |
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Pages (from-to) | 155-183 |

Number of pages | 29 |

Journal | Linear Algebra and its Applications |

Volume | 697 |

Early online date | 13 Oct 2023 |

DOIs | |

Publication status | Published - 15 Sept 2024 |

### Bibliographical note

Publisher Copyright:© 2023 The Author(s)

## Keywords

- Fredholm characteristics
- Rational matrix functions
- Toeplitz kernels
- Toeplitz operators
- Unbounded operators