Spherical Tuples of Hilbert Space Operators

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 <= 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 < infinity.