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

2016/05/17 by Gabriel Larotonda, Larotonda, Gabriel · 1 citation
Mathematics · #15A45 (Primary) #47A30 #47A63 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:15A45 #msc:47A30 #msc:47A63

paper · pdf · doi:10.48550/arxiv.1605.05377

3 pages (shortened). Added references and minor corrections

arxiv created 2016/10/04 · arxiv updated 2016/10/06

Abstract

Let p>1 and 1/p+1/q=1. Consider Hölder's inequality ‖ab^*‖₁≤ ‖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≥ 0.

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