N-hypercontractivity and similarity of Cowen-Douglas operators

Abstract

When the backward shift operator on a weighted space Hw2={f=∑j=0∞ajzj:∑j=0∞|aj|2wj<∞} is an n-hypercontraction, we prove that the weights must satisfy the inequalitywj+1wj≤1+jn+j. As an application of this result, it is shown that such an operator cannot be subnormal. We also give an example to illustrate the important role that the n-hypercontractivity assumption plays in determining the similarity of Cowen-Douglas operators in terms of the curvatures of their eigenvector bundles.