Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II

Abstract

© 2020, Springer Nature Switzerland AG. Let M be a von Neumann algebra of operators on a Hilbert space H and τ be a faithful normal semifinite trace on M. Let tτ be the measure topology on the ∗ -algebra S(M, τ) of all τ-measurable operators. We define three tτ-closed classes P1, P2 and P3 of τ-measurable operators and investigate their properties. The class P2 contains P1∪ P3. If a τ-measurable operator T is hyponormal, then T lies in P1∩ P3; if an operator T lies in P3, then UTU∗ belongs to P3 for all isometries U from M. If a bounded operator T lies in P1∪ P3 then T is normaloid. If an operator T∈ S(M, τ) is p-hyponormal with 0 < p≤ 1 then T∈ P1. If M= B(H) and τ=tr is the canonical trace, then the class P1 (resp., P3) coincides with the set of all paranormal (resp., ∗ -paranormal) operators on H. Let A, B∈ S(M, τ) and A be p-hyponormal with 0 < p≤ 1. If AB is τ-compact then A∗B is τ-compact.