$N$-hypercontractivity and similarity of Cowen-Douglas operators

When the backward shift operator on a weighted space $H^2_w={f=\sum_{j=0} ^{\infty} a_jz^j : \sum_{j=0}^{\infty} |a_j|^2w_j < \infty}$isan$n$-hypercontraction,weprovethattheweightsmustsatisfytheinequality$$\frac{w_{j+1}}{w_j}\le \frac{1+j}{n+j}\mathrm{.}$$ 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.