In this paper, we are concerned with hyponormality and subnormality of block
Toeplitz operators acting on the vector-valued Hardy space HCn2 of the unit
circle. First, we establish a tractable and explicit criterion on the
hyponormality of block Toeplitz operators having bounded type symbols via the
triangularization theorem for compressions of the shift operator. Second, we
consider the gap between hyponormality and subnormality for block Toeplitz
operators. This is closely related to Halmos's Problem5: Is every subnormal
Toeplitz operator either normal or analytic? We show that if Φ is a
matrix-valued rational function whose co-analytic part has a coprime
factorization then every hyponormal Toeplitz operator T Φ whose
square is also hyponormal must be either normal or analytic. Third, using the
subnormal theory of block Toeplitz operators, we give an answer to the following
"Toeplitz completion" problem: find the unspecified Toeplitz entries of the
partial block Toeplitz matrix A= so that A becomes subnormal, where U is the
unilateral shift on H 2.