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arXiv · 1605.05377

The case of equality in Hölder's inequality for matrices and operators

Gabriel Larotonda
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Abstract

Let $p>1$ and $1/p+1/q=1$. Consider Hölder'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 $|a|^p=λ|b|^q$ for some $λ\ge 0$.

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Gabriel Larotonda. 2016-10-04. The case of equality in Hölder's inequality for matrices and operators. https://arxiv.org/abs/1605.05377

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