Which subnormal Toeplitz operators are either normal or analytic ?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmos’s Problem 5: Which subnormal block Toeplitz operators are either normal or analytic ? We extend and prove Abrahamse’s Theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factorization,” is normal or analytic. We also prove that the left coprime factorization condition is essential. Finally, we examine a well known conjecture, of whether every submormal Toeplitz operator with finite rank selfcommutator is normal or analytic.