Subnormal block Toeplitz operators

In this paper we consider the subnormality of block Toeplitz operators $T_{\mathrm{\Phi }}$, where $\mathrm{\Phi }$ is an $n\times n$ matrix-valued function on the unit circle $\mathbb{T}$ of the form $$\mathrm{\Phi }=Q{\mathrm{\Phi }}^{\ast }\text{(}Q\text{ is a finite Blaschke–Potapov product).}$$ This is related to a matrix-valued version of Halmos’s Problem 5 and Nakazi-Takahashi Theorem. We ask whether $T_{\mathrm{\Phi }}$ is either normal or analytic if $T_{\mathrm{\Phi }}$ is subnormal, where $\mathrm{\Phi }$ 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 ${\mathrm{\Phi }}^{\ast }$ is not of bounded type.