# Journal of Operator Theory

Volume 81, Issue 1, Winter 2019 pp. 157-173.

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

**Authors**:
Gabriel Larotonda

**Author institution:** Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina, \textit{and}
Instituto Argentino de Matem\'atica ``Alberto P. Calder\'on'', CONICET, Buenos Aires, Argentina

**Summary: ** 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.

**DOI: **http://dx.doi.org/10.7900/jot.2017dec15.2182

**Keywords: ** measurable operator, $\tau$-compact operator, semi-finite von Neumann algebra, Young's inequality, $s$-number

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