Abrahamse's Theorem for matrix-valued symbols and subnormal Toeplitz completions

Abstract

This paper deals with subnormality of Toeplitz operators with matrix-valued symbols and, in particular, with an appropriate reformulation of Halmos's Problem 5: Which subnormal Toeplitz operators with matrix-valued symbols are either normal or analytic? In 1976, M. Abrahamse showed that if $\varphi\in L^\infty$ is such that $\varphi$ or $\overline\varphi$ is of bounded type and if $T_\varphi$ is subnormal, then $T_\varphi$ is either normal or analytic. In this paper we establish a matrix-valued version of Abrahamse's Theorem and then apply this result to solve the following Toeplitz completion problem: Find the unspecified Toeplitz entries of the partial block Toeplitz matrix $$ A:=\begin{bmatrix} T_{\overline b_\alpha} & ?\\?& T_{\overline b_\beta}\end{bmatrix}\quad\hbox{($\alpha,\beta\in\mathbb D$)} $$ so that $A$ becomes subnormal, where $b_\lambda$ is a Blaschke factor of the form $b_\lambda(z):=\frac{z-\lambda}{1-\overline \lambda z}$ ($\lambda\in \mathbb D$).