Young's (in)equality for compact operators

2015/05/09 by Larotonda, Gabriel
#15A45 #47A30 (Primary) 15A42 #47A63 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1505.02267

Abstract

If a,b are n× n matrices, Ando proved that Young's inequality is valid for their singular values: if p>1 and 1/p+1/q=1, then λ_k|ab^*|≤ λ_k( \frac1p |a|^p+\frac 1q |b|^q ) for all k. Later, this result was extended for the singular values of a pair of compact operators acting on a Hilbert space by Erlijman, Farenick and Zeng. In this paper we prove that if a,b are compact operators, then equality holds in Young's inequality if and only if |a|^p=|b|^q, obtaining a complete characterization of such a,b in relation to other (operator norm) Young inequalities.

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