Hyponormality of bounded-type Toeplitz operators (Q2922197)

Let \(L^2\) be the Lebesgue space over the unit circle \(\mathbf T\) in the complex plane \(\mathbb C\) with respect to the normalized arclength measure and \(H^2\) be the corresponding Hardy space. Given a positive integer \(n\), let \(P_n\) be the Hilbert space orthogonal projection from \(L^2\otimes {\mathbb C}^n\) onto \(H^2\otimes {\mathbb C}^n\). Given \(\Phi=(\varphi_{ij})_{n\times n}\), where each \(\varphi_{ij}\) is an essentially bounded measurable function on \(\mathbb T\), the block Toeplitz operator \(T_\Phi\) with symbol \(\Phi\) is defined by \(T_\Phi f:= P_n(\Phi f)\) for \(f\in H^2\otimes {\mathbb C}^n\). The symbol \(\Phi\) is said to be of bounded type if each entry function is a quotient of two bounded analytic functions.NEWLINENEWLINEIn this paper, the authors obtain a hyponormality criterion for operators \(T_\Phi\) when \(\Phi\) is a normal (matrix-valued) function such that \(\Phi\) and \(\Phi^*\) are of bounded type. They also obtain a reduced criterion when \(\Phi\) is a normal rational function. The criteria are described by means of terms involving the right coprime factorizations of the projections of the symbol \(\Phi\). Various examples are provided to illustrate the criteria.