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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.

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Acknowledgments

Authors wish to place on record their sincere thanks to the referee for pointing out a couple of careless assesrtions in the original manuscript and also for a number of valuable suggestions for the improvement of the presentation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, Kanpur, 208016, India

    Sameer Chavan

  2. Center for Postgraduate Studies in Mathematics, S. P. College, Pune, 411030, India

    V. M. Sholapurkar

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  1. Sameer Chavan
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  2. V. M. Sholapurkar
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Correspondence to V. M. Sholapurkar.

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Communicated by Mihai Putinar.

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Chavan, S., Sholapurkar, V.M. Rigidity Theorems for Spherical Hyperexpansions. Complex Anal. Oper. Theory 7, 1545–1568 (2013). https://doi.org/10.1007/s11785-012-0270-6

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  • DOI: https://doi.org/10.1007/s11785-012-0270-6

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