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On noncommutative Hölder inequality of Sukochev and Zanin for weak Schatten class

Yi C. Huang, Sijie Luo

Abstract

Sukochev and Zanin resolved an open problem due to B. Simon concerning optimal constants in Hölder inequality for the weak Schatten classes of compact operators. In this note we observe that these constants, by introducing the modified weak Schatten quasi-norms, can be renormalised so that the original Simon's conjecture (with optimal constant 1) does hold. We also provide an unexpectedly simple proof for the modified Hölder inequality and its sharpness.

On noncommutative Hölder inequality of Sukochev and Zanin for weak Schatten class

Abstract

Sukochev and Zanin resolved an open problem due to B. Simon concerning optimal constants in Hölder inequality for the weak Schatten classes of compact operators. In this note we observe that these constants, by introducing the modified weak Schatten quasi-norms, can be renormalised so that the original Simon's conjecture (with optimal constant 1) does hold. We also provide an unexpectedly simple proof for the modified Hölder inequality and its sharpness.
Paper Structure (2 sections, 2 theorems, 9 equations)

This paper contains 2 sections, 2 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:FZ']}

Key Result

Theorem 1.3

Let $p, q, r>0$ be such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Then and the constant is optimal.

Theorems & Definitions (4)

  • Definition 1.1
  • Conjecture 1.2
  • Theorem 1.3: Sukochev-Zanin
  • Theorem 1.4: $\simeq$ Sukochev-Zanin