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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