Subnormal and hyponormal generators of C*-algebras

For a Hilbert space operator A, that is not a normal operator, we give some necessary conditions on the thickness of the spectrum of A for C*(A) to have a subnormal generator. When A is irreducible and essentially normal, whether or not C*(A) has a subnormal generator depends only on the spectral picture of A. We show that under certain conditions on the essential spectrum and Fredholm index function of A that C*(A) has a subnormal or a hyponormal generator.