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 Sp-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 Sp 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 Sp 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 Sp for any p < ∞.

Commuting tuple of multiplication operators homogeneous under the unitary group

“…This is Proposition 2.6. For the proof, we need the following elementary lemma (compare with Lemma 2.8 of [5]). Lemma 2.5.…”

“…This is Proposition 2.6. For the proof, we need the following elementary lemma (compare with Lemma 2.8 of [5]). Lemma Let $$\mathcal {H}_1:=(\mathcal {H}, \langle \cdot, \cdot \rangle _1)$$ and $$\mathcal {H}_2:=(\mathcal {H}, \langle \cdot, \cdot \rangle _2)$$ be two Hilbert spaces.…”

“…In particular, when $$\Omega$$ is the Euclidean ball $$\mathbb {B}_d$$ in $$\mathbb {C}^d$$, then $$\mathbb {K}$$ is the group of unitary linear transformations on $$\mathbb {C}^d$$ and the spherical tuples defined in [5] are nothing but $$\mathcal {U}(d)$$‐homogeneous $$d$$‐tuples. In this paper, we would be discussing $$\mathcal {U}(d)$$‐homogeneous commuting $$d$$‐tuple $$\bm{M}$$ of multiplication by coordinate functions $$z_1,\ldots , z_d$$ on a reproducing kernel Hilbert space $$\mathcal {H}_K(\mathbb {B}_d, \mathbb {C}^n)$$.…”

“…In particular, when Ω is the Euclidean ball 𝔹 𝑑 in ℂ 𝑑 , then 𝕂 is the group of unitary linear transformations on ℂ 𝑑 and the spherical tuples defined in [5] are nothing but  (𝑑)-homogeneous 𝑑-tuples. In this paper, we would be discussing  (𝑑)-homogeneous commuting 𝑑-tuple 𝑴 of multiplication by coordinate functions 𝑧 1 , … , 𝑧 𝑑 on a reproducing kernel Hilbert space  𝐾 (𝔹 𝑑 , ℂ 𝑛 ).…”