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


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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