On the $N$-hypercontractions and similarity of multivariable weighted shifts

TL;DR

This paper extends Shields' theorem to multivariable weighted shifts, providing criteria for their similarity and hypercontractivity using bundle curvature and weight sequences.

Contribution

It generalizes the similarity theorem for unilateral shifts to multivariable cases and links hypercontractivity to weight sequences.

Findings

Necessary and sufficient conditions for similarity of multivariable weighted shifts.

A necessary condition for hypercontractivity based on weight sequences.

Application of bundle curvature in operator similarity analysis.

Abstract

In \cite{SH}, A. L. Shields proved a well-known theorem for the similarity of unilateral weighted shift operators. By using the generalization of this theorem for multivariable weighted shifts and the curvature of holomorphic bundles, we give a necessary and sufficient condition for the similarity of $m$-tuples in Cowen-Douglas class. We also present a necessary condition for commuting $m$-tuples of backward weighted shift operators to be $n$-hypercontractive in terms of the weight sequences.