Young's (in)equality for compact operators

Abstract

If a, b are n × n matrices, T. Ando proved that Young’s inequality is valid for their singular values: if p > 1 and 1/p + 1/q = 1, then λk(|ab∗ |) ≤ λk 1 p |a| p + 1 q |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 J. Erlijman, D. Farenick and R. 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 .