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Generalized s-numbers of $$\tau$$-measurable operators. (English) Zbl 0617.46063
Let M be a semi-finite von Neumann algebra and $$\tau$$ a faithful trace of M. A densely defined closed oprator T affiliated with M is said to be $$\tau$$-measurable if for each $$\epsilon >0$$ there exists a projection E in M such that $$E({\mathcal H})\subseteq$$ domain of and $$\tau(I-E)\leq \epsilon$$. Let T be $$\tau$$-measurable operator and $$t>0$$. The $$t^{th}$$ singular number of T is $u_ t(T):\inf \{\| TE\|:E\text{ a projection in M, }\tau(I-e)\leq t\}.$ This is an expository article on singular numbers of measurable operators. Applications to dominated convergence theorems and convexity inequalities are investigated.
Reviewer: S.Sankaran

##### MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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