, Volume 7, Issue 5, pp 1545-1568
Date: 27 Oct 2012

Rigidity Theorems for Spherical Hyperexpansions

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Abstract

The class of spherical hyperexpansions is a multi-variable analog of the class of hyperexpansive operators with spherical isometries and spherical 2-isometries being special subclasses. It is known that in dimension one, an invertible \(2\) -hyperexpansion is unitary. This rigidity theorem allows one to prove a variant of the Berger–Shaw Theorem which states that a finitely multi-cyclic \(2\) -hyperexpansion is essentially normal. In the present paper, we seek for multi-variable manifestations of this rigidity theorem. In particular, we provide several conditions on a spherical hyperexpansion which ensure it to be a spherical isometry. We further carry out the analysis of the rigidity theorems at the Calkin algebra level and obtain some conditions for essential normality of a spherical hyperexpansion. In the process, we construct several interesting examples of spherical hyperexpansions which are structurally different from the Drury-Arveson \(m\) -shift.

Communicated by Mihai Putinar.