  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 analytic functions), whose co-analytic part has
a "coprime decomposition," is normal or analytic. We also prove that the
coprime decomposition condition is essential. Finally, we examine a well known
conjecture, of whether every submormal Toeplitz operator with finite rank
self-commutator is normal or analytic.
