Integrable products of measurable operators

Abstract

© 2016, Pleiades Publishing, Ltd.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 M is the *-algebra of all τ-measurable operators. We give the sufficient conditions for integrability of operator product A,B 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 ∈ M cannot be nilpotent. If A ∈ 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 ∈ M; is such that |A| log+|A|, ep|B| ∈ L1(M, τ) then AB,BA ∈ L1(M, τ).