k-Hyponormality and n-Contractivity for Agler-Type Shifts

We consider k-hyponormality and n-contractivity (k, n = 1, 2, ...) as "weak subnormalities" for a Hilbert space operator. It is known that k-hyponormality implies 2k-contractivity; we produce some classes of weighted shifts including a parameter for which membership in a certain n-contractive class is equivalent to k-hyponormality. We consider as well some extensions of these results to operators arising as restrictions of these shifts, or from linear combinations of the Berger measures associated with the shifts.