Weakly $\mathcal U(d)$-homogeneous commuting tuple of bounded operators

TL;DR

This paper introduces the concept of weakly $$-homogeneous commuting operator tuples, providing conditions for similarity to $$-homogeneous tuples and characterizing weakly $$-homogeneous multishifts, especially spherically balanced ones.

Contribution

It defines weakly $$-homogeneous tuples, establishes similarity conditions, and fully characterizes weakly $$-homogeneous multishifts, including a refinement for spherically balanced cases.

Findings

Weakly $$-homogeneous tuples can be similar to $$-homogeneous tuples under certain conditions.

A multishift is weakly $$-homogeneous if and only if it is similar to a $$-homogeneous multishift.

Characterization of weakly $$-homogeneous multishifts is refined for spherically balanced multishifts.

Abstract

We introduce and study the weakly $\mathcal U(d)$-homogeneous commuting tuple of operators. We provide a sufficient condition under which a weakly $\mathcal U(d)$-homogeneous tuple is similar to a $\mathcal U(d)$-homogeneous tuple. Further, we focus our attention to multishifts and completely characterize weakly $\mathcal U(d)$-homogeneous multishifts. In particular, we show that a multishift is weakly $\mathcal U(d)$-homogeneous if and only if it similar to a $\mathcal U(d)$-homogeneous multishift. The results for multishifts are further refined for the class of spherically balanced multishifts.