Some block Toeplitz composition operators

The composition operators on H2(D)H2(D) with minimal polynomial zn−1zn−1 are shown to be block Toeplitz with Toeplitz symbol equal to an n×nn×n matrix-valued polynomial of degree 1. This result is used to prove that the numerical range of a composition operator on H2(D)H2(D) with minimal polynomial z3−1z3−1 cannot be a circular disk.