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 $p <\infty$.