On inequalities involving the spherical operator transforms

TL;DR

The paper advances the multivariable operator theory of spherical transforms by establishing refinements and extensions of norm inequalities for operator tuples. It introduces and analyzes the spherical Aluthge, Duggal, mean, and Heinz transforms within the joint setting, proving tight relations among spherical norms, joint numerical radii, and their Schatten-p counterparts. Key contributions include sharp multivariable Heinz-type bounds, bounds for spherical Schatten $p$-norms under transforms, and foundational inequalities for the joint Schatten $p$-numerical radius and Schatten hypo-$p$-norm, along with equality characterizations linked to normality. The results deepen the understanding of how spherical transforms interact with unitarily invariant norms in the multivariable context and provide tools for applications in multivariate operator theory and related areas.

Abstract

This paper explores refinements of some operator norm inequalities through the generalized spherical Aluthge transform and the spherical Heinz transform. We introduce the spherical Schatten $p$-norm for operator tuples and establish several related inequalities. Additionally, equality conditions for some of these inequalities are also presented. Furthermore, we define the (joint) Schatten $p$-numerical radius and the Schatten hypo-$p$-norm for operator tuples, deriving some fundamental inequalities in this setting.

Theorems & Definitions (51)

• Lemma 1.1
• Lemma 1.2
• Theorem 1.1
• Definition 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof