Spherical Tuples of Hilbert Space Operators

Abstract

We introduce and study a class of operator tuples in complex Hilbert spaces, which we call spherical tuples. In particular, we characterize spherical multi-shifts, and more generally, multiplication tuples on RKHS. We further use these characterizations to describe various spectral parts including the Taylor spectrum. We also find a criterion for the Schatten $S_p$-class membership of cross-commutators of spherical $m$-shifts. We show, in particular, that cross-commutators of non-compact spherical $m$-shifts cannot belong to $S_p$ for $p\le m$. We specialize our results to some well-studied classes of multi-shifts. We prove that the cross-commutators of a spherical joint $m$-shift, which is a $q$-isometry or a $2$-expansion, belongs to $S_p$ if and only if $p>m$. We further give an example of a spherical jointly hyponormal $2$-shift, for which the cross-commutators are compact but not in $S_p$ for any $p<\infty$.

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References in corpus (3)

• p-Summable Commutators in Dimension d
• Essentially Reductive Weighted Shift Hilbert Modules
• Spectral pictures of 2-variable weighted shifts

Cited by in corpus (1)

• An Analytic Model for Left-Invertible Weighted Shifts on Directed Trees