Subnormal block Toeplitz operators

In this paper we consider the subnormality of block Toeplitz operators TΦ, where Φ is an n × n matrix-valued function on the unit circle $$\mathbb{T}$$ of the form $$\Phi=Q\Phi^{\ast}\;\;\;\;(Q\;\text{is}\; \text{a}\; \text{finite}\; \text{Blaschke-Potapov}\; \text{product})$$ This is related to a matrix-valued version of Halmos’ Problem 5 and the Nakazi–Takahashi Theorem. We ask whether TΦ is either normal or analytic if TΦ is subnormal, where Φ is of the above form. We give answers to this problem for different cases of the symbol. Moreover, we provide a sufficient condition for the answer to be affirmative when Φ* is not of bounded type.