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

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.