Young's (in)equality for compact operators

Abstract

If $a,b$ are $n\times n$ matrices, Ando proved that Young's inequality is valid for their singular values: if $p>1$ and $1/p+1/q=1$, then $$ \lambda_k|ab^*|\le \lambda_k( \frac1p |a|^p+\frac 1q |b|^q ) \, \textit{ for all }k. $$ 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.