A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm characteristics (Q6565839)

Let \(\mathbb D\) (resp. \(\overline{\mathbb D}\)) be the open unit complex disk (resp. the closed of \(\mathbb D\)), \(\text{Rat}^{m\times n}(\mathbb T)\) be the space of \(m\times n\) rational matrix functions with poles on \(\mathbb T\), the unit complex circle. Let \(\Omega(z)\in \text{Rat}^{m\times n}(\mathbb T)\) satisfying the Wiener-Hopf factorization, i.e., \(\Omega(z)=z^{-k}\Omega_-(z)\Omega_0(z)P_0(z)\Omega_+(z)\) for some \(k\ge 0\), where \(\Omega_-\) (resp. \(\Omega_+\)) is an element of \(\text{Rat}^{m\times n}(\mathbb T)\) such that it has no poles outside (resp. in) of \(\mathbb D\) (resp. \(\overline{\mathbb D}\)), \(\Omega_0=Diag_{j=1}^m(\varphi_j)\) where each \(\varphi_j\) is a fractional function on \(\mathbb T\) with zeros and poles on \(\mathbb T\), and \(P_0(z)\) is the lower triangular with \(\det(P_0(z))=z^N\) for some \(N\in\mathbb N\). Let \(L^p_m(\mathbb T)\) (resp. \(H_m^p(\mathbb T)\)) be the Banach space comprised by vector functions with \(m\)-components and each one belongs to \(L^p(\mathbb T)\), the classical \(p\)-integrable space on \(\mathbb T\). Then the authors define \(T_\Omega\), the Toeplitz-like operator with symbol \(\Omega(z)\in\text{Rat}^{m\times n}(\mathbb T)\) such that the determinant of \(\Omega(z)\) does not vanish and the domain of \(T_\Omega\) is defined as the set of \(f\in H_m^p\) such that \(\Omega f=h+\eta\) with \(h\in L_m^p(\mathbb T)\) and \(\eta\) is a vector fractional function with \(m\)-components and each one has poles and zeros on \(\mathbb T\) and \(T_\Omega f=\mathbf{P}h\) where \(\mathbf{P}\) represents the vector projection operator from \(L_m^p(\mathbb T)\) onto \(H_m^p(\mathbb T)\). Then, by assuming a meaningful assumption on the Wiener-Hopf type factorization, the authors state a formula for the dimension of the kernel of \(T_\Omega\), see Theorem 1.4 for the full statement.