SAMS Subject Classification Number: 11

Let \(\Omega\in{\mathrm{Rat}}^{m\times m}\) with possibly poles on \({\mathbb T}\), where \({\mathrm{Rat}}^{m\times n}\) the space of \(m \times n\) rational matrix functions and define the Toeplitz-like operator \(T_\Omega\left (H^p_m \rightarrow H^p_m \right )\) as follows: \[{\mathrm{Dom}\, }(T_\Omega)=\left\{\begin{array}{ll} f\in H^p_m :& \begin{array}{l} \Omega f = h + r \textrm{ where } h\in L^p_m({\mathbb T}),\\ \quad\mbox{and}\quad r \in{\mathrm{Rat}}_0^{m}({\mathbb T}) \\ \end{array}\\ \end{array} \right \}\qquad{(1)}\] \[T_\Omega f = {\mathbb P}h \textrm{ where } {\mathbb P}\textrm{ is the Riesz projection of } L^p_m({\mathbb T}) \textrm{ onto }H^p_m.\qquad{(2)}\] This talk will discuss some basic properties of the operator \(T_\Omega\) such as

1. \(T_\Omega\) is a densely defined closed operator and its domain contains the polynomials, and

2. \(T_\Omega\) satisfies a Brown-Halmos type condition on its domain.

as well as Fredholm properties. The construction of a Wiener-Hopf-type factorization of \(\Omega\), where \(\Omega\) has poles on \({\mathbb T}\), is shown to allow one to determine the Fredholm properties of the operator \(T_\Omega\). In addition, we will contrast some properties with the scalar case as well as \(T_\Omega\) with classical block Toeplitz operators.