Integrable products of measurable operators

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) ⇔ AB ∈ L_p(M, τ) ⇔ AB* ∈ 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(Aⁿ) = 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| ∈ L₁(M, τ) then AB,BA ∈ L₁(M, τ).