$T$ is an operator in $M$, $M$ is finite von Neumann algebra. There is a notion of singular value function that is ($s$-numbers). My question is: what is $s$-number for tensor product of two operators and how they are related to their individual's s-numbers? $s$-number is defined by $\mu_{t}(T)=\underset{E \text{ is a projection in }M \text{ with }\tau(I-E)\leq t}\inf[\underset{\xi \in E(\mathcal{H}),\|\xi\|=1 }\sup \langle T\xi,\xi\rangle]$, $t$ in $(0,1)$. For more equivalent definitions one can look at Fack's paper on $s$ numbers for $\tau$ measurable operators and Dykemma's paper on brown measures too.
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$\begingroup$ The question might be improved if you could quickly define what the $s$-numbers are. And perhaps provide a good reference for further details? $\endgroup$– Matthew DawsCommented Mar 15, 2019 at 13:45
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3$\begingroup$ @MatthewDaws: See, for instance, “Generalized s-numbers of τ-measurable operators” by Thierry Fack and Hideki Kosaki, projecteuclid.org/euclid.pjm/1102701004. $\endgroup$– Dmitri PavlovCommented Mar 15, 2019 at 13:48
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$\begingroup$ you see in Fack s paper as Pavlov mentioned $\endgroup$– mathloverCommented Mar 15, 2019 at 14:45
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$\begingroup$ Edited @Matthew Daws, please have a look. $\endgroup$– user136400Commented Mar 16, 2019 at 7:12
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