Abstract
Let τ be a faithful normal semifinite trace on von Neumann algebra M, 0 < p < +∞ and L p (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 ∈ L p (M, τ) ⇔ A B ∈ L p (M, τ) ⇔ A B* ∈ L p (M, τ); moreover, ||AB|| p = |||A|B|| p = |||A||B*||| p . If A is hyponormal, B is cohyponormal and AB ∈ L p (M, τ) then BA ∈ L p (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 μ t (A n) = μ t (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|, e p|B| ∈ L 1(M, τ) then AB,BA ∈ L 1(M, τ).
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Submitted by Oleg Tikhonov
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Bikchentaev, A. Integrable products of measurable operators. Lobachevskii J Math 37, 397–403 (2016). https://doi.org/10.1134/S1995080216040041
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DOI: https://doi.org/10.1134/S1995080216040041