On n-contractive and n-hypercontractive Operators

We consider in this paper the classes of n-hypercontractive Hilbert space operators, primarily weighted shifts, and obtain results for back step extensions of recursively generated subnormal weighted shifts and for perturbations in the first weight of the Bergman shift. We compare the results with those for the classes of k-hyponormal operators, and recapture, by an n-hypercontractive approach, a subnormality result originally proved in the k-hyponormal context.