Paper detail

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

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 $φ\in L^\infty$ is such that $φ$ or $\overlineφ$ is of bounded type and if $T_φ$ is subnormal, then $T_φ$ 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 $$ A:=\begin{bmatrix} T_{\overline b_α} & ?\\?& T_{\overline b_β}\end{bmatrix}\quad\hbox{($α,β\in\mathbb D$)} $$ so that $A$ becomes subnormal, where $b_λ$ is a Blaschke factor of the form $b_λ(z):=\frac{z-λ}{1-\overline λz}$ ($λ\in \mathbb D$).