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) ⇔ 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).