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.