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arXiv · 1901.09471

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

Kui Ji·Hyun-Kyoung Kwon·Jing Xu
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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.

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Kui Ji, Hyun-Kyoung Kwon, Jing Xu. 2019-01-28. $N$-hypercontractivity and similarity of Cowen-Douglas operators. https://arxiv.org/abs/1901.09471

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