# On $s$-numbers in finite von Neumann algebra

$$T$$ 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.

• The question might be improved if you could quickly define what the $s$-numbers are. And perhaps provide a good reference for further details? – Matthew Daws Mar 15 at 13:45
• @MatthewDaws: See, for instance, “Generalized s-numbers of τ-measurable operators” by Thierry Fack and Hideki Kosaki, projecteuclid.org/euclid.pjm/1102701004. – Dmitri Pavlov Mar 15 at 13:48
• you see in Fack s paper as Pavlov mentioned – mathlover Mar 15 at 14:45
• Edited @Matthew Daws, please have a look. – user136400 Mar 16 at 7:12