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

In a recent paper (Groenewald et al.\ {\em Complex Anal.\ Oper.\ Theory} \textbf{15:1} (2021)) we considered an unbounded Toeplitz-like operator $T_\Omega$ generated by a rational matrix function $\Omega$ that has poles on the unit circle $\mathbb{T}$ of the complex plane. A Wiener-Hopf type factorization was proved and this factorization was used to determine some Fredholm properties of the operator $T_\Omega$, 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 $T_\Omega$ from this factorization, and hence of the co-kernel, even when $T_\Omega$ is Fredholm. In the current paper we provide a formula for the dimension of the kernel of $T_\Omega$ under an additional assumption on the Wiener-Hopf type factorization. In the case that $\Omega$ is a $2 \times 2$ 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 $2 \times 2$ case is partially extended to the case of matrix functions of arbitrary size.