$T$ 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.

  • $\begingroup$ 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 Daws Mar 15 at 13:45
  • 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 Pavlov Mar 15 at 13:48
  • $\begingroup$ you see in Fack s paper as Pavlov mentioned $\endgroup$ – mathlover Mar 15 at 14:45
  • $\begingroup$ Edited @Matthew Daws, please have a look. $\endgroup$ – user136400 Mar 16 at 7:12

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