The case of equality in H\"older's inequality for matrices and operators

TL;DR

This paper provides a simple proof characterizing when equality holds in H"older's inequality for matrices and operators, showing it occurs precisely when the p-th power of one operator is proportional to the q-th power of the other.

Contribution

It offers a straightforward proof of the equality condition in H"older's inequality for various operator classes, simplifying previous approaches.

Findings

Equality in H"older's inequality occurs iff |a|^p = λ|b|^q for some λ ≥ 0.

The proof is based on the case p=2, simplifying the understanding of equality conditions.

Applicable to matrices, compact operators, and elements of finite C*-algebras or semi-finite von Neumann algebras.

Abstract

Let $p>1$ and $1/p+1/q=1$. Consider H\"older's inequality $$ \|ab^*\|_1\le \|a\|_p\|b\|_q $$ for the $p$-norms of some trace ($a,b$ are matrices, compact operators, elements of a finite $C^*$-algebra or a semi-finite von Neumann algebra). This note contains a simple proof (based on the case $p=2$) of the fact that equality holds iff $|a|^p=\lambda |b|^q$ for some $\lambda\ge 0$.