Two classes of τ-measurable operators affiliated with a von Neumann algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P 1 and P 2 of τ-measurable operators and investigate their properties. The class P 2 contains P 1. If a τ-measurable operator T is hyponormal, then T lies in P 1; if an operator T lies in P k , then UTU* belongs to P k for all isometries U from M and k = 1, 2; if an operator T from P 1 admits the bounded inverse T −1, then T −1 lies in P 1. We establish some new inequalities for rearrangements of operators from P 1. If a τ-measurable operator T is hyponormal and T n is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P 1 coincides with the set of all paranormal operators on H.