Integrable products of measurable operators
- Authors: Bikchentaev A.1
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Affiliations:
- Kazan (Volga Region) Federal University
- Issue: Vol 37, No 4 (2016)
- Pages: 397-403
- Section: Article
- URL: https://journals.rcsi.science/1995-0802/article/view/197991
- DOI: https://doi.org/10.1134/S1995080216040041
- ID: 197991
Cite item
Abstract
Let τ be a faithful normal semifinite trace on von Neumann algebra M, 0 < p < +∞ and Lp(M, τ) be the space of all integrable (with respect to τ) with degree p operators, assume also that \(\widetilde M\) is the *-algebra of all τ-measurable operators. We give the sufficient conditions for integrability of operator product \(A,\;B \in \widetilde M\). We prove that AB ∈ Lp(M, τ) ⇔ AB ∈ Lp(M, τ) ⇔ AB* ∈ Lp(M, τ); moreover, ||AB||p = |||A|B||p = |||A||B*|||p. If A is hyponormal, B is cohyponormal and AB ∈ Lp(M, τ) then BA ∈ Lp(M, τ) and ||BA||p ≤ ||AB||p; for p = 1 we have τ(AB) = τ(BA). A nonzero hyponormal (or cohyponormal) operator \(A \in \widetilde M\) cannot be nilpotent. If \(A \in \widetilde M\) is quasinormal then the arrangement μt(An) = μt(A)n for all n ∈ N and t > 0. If A is a τ-compact operator and \(B \in \widetilde M\) is such that |A| log+|A|, ep|B| ∈ L1(M, τ) then AB,BA ∈ L1(M, τ).
About the authors
A. Bikchentaev
Kazan (Volga Region) Federal University
Author for correspondence.
Email: Airat.Bikchentaev@kpfu.ru
Russian Federation, Kremlevskaya ul. 18, Kazan, Tatarstan, 420008