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

## TL;DR

This paper investigates the conditions for equality in Young's inequality for s-numbers within semi-finite von Neumann algebras, establishing that equality implies a specific power relation between the operators.

## Contribution

It provides a characterization of equality cases in Young's inequality for s-numbers in semi-finite von Neumann algebras, extending results to unbounded operators.

## Key findings

- Equality in Young's inequality occurs only when |a|^p=|b|^q.
- The results extend to unbounded operators affiliated with the algebra.
- The work relates to other symmetric norm Young inequalities.

## Abstract

For a semi-finite von Neumann algebra $\mathcal A$, we study the case of equality in Young's inequality of s-numbers for a pair of $\tau$-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 $\mathcal A$, and relate this problem with other symmetric norm Young inequalities.