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

## TL;DR

This paper investigates the properties of backward shift operators on weighted spaces under hypercontractivity conditions, revealing restrictions on weights, implications for subnormality, and their influence on the similarity of Cowen-Douglas operators.

## Contribution

It establishes a new inequality relating weights and hypercontractivity, and demonstrates how hypercontractivity affects the similarity classification of Cowen-Douglas operators.

## Key findings

- Weights must satisfy a specific ratio inequality under hypercontractivity.
- Such operators cannot be subnormal.
- Hypercontractivity influences the similarity of Cowen-Douglas operators via curvature.

## Abstract

When the backward shift operator on a weighted space $H^2_w=\{f=\sum_{j=0} ^{\infty} a_jz^j : \sum_{j=0}^{\infty} |a_j|^2w_j < \infty\}$ is an $n$-hypercontraction, we prove that the weights must satisfy the inequality $$\frac{w_{j+1}}{w_j} \leq {\frac{1+j}{n+j}}.$$ 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.