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 $\varphi\in L^\infty$issuchthat$\varphi$or$\overline\varphi$ is of bounded type and if $T_{\phi }$ is subnormal, then $T_{\phi }$ 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 $$A:=\left[\begin{array}{cc}{T}_{{\bar{b}}_{\alpha }}& ?\\ ?& T_{{\bar{b}}_{\beta }}\end{array}\right]\text{(}\alpha ,\beta \in \mathbb{D}\text{)}$$ so that $A$ becomes subnormal, where $b_{\lambda }$ is a Blaschke factor of the form $b_{\lambda }(z):=\frac{z-\lambda }{1-\bar{\lambda }z}$ ($\lambda\in \mathbb D$).