Young's (in)equality for compact operators

TL;DR

This paper extends Young's inequality for singular values from finite matrices to compact operators on Hilbert spaces, providing a complete characterization of cases where equality holds.

Contribution

It proves that equality in Young's inequality for compact operators occurs if and only if specific operator power relations hold, generalizing previous finite-dimensional results.

Findings

Equality in Young's inequality for compact operators is characterized by the condition |a|^p=|b|^q.

The result generalizes finite-dimensional matrix inequalities to infinite-dimensional operators.

Provides a complete characterization of equality cases in operator Young's inequality.

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.