The case of equality in Young's inequality for the s-numbers in semi-finite von Neumann algebras

Abstract

For a semi-finite von Neumann algebra A, we study the case of equality in Young's inequality of s-numbers for a pair of τ-measurable operators a,b, and we prove that equality is only possible if |a|p=|b|q. We also extend the result to unbounded operators affiliated with A, and relate this problem with other symmetric norm Young inequalities.