Paranormal Measurable Operators Affiliated with a Semifinite von Neumann Algebra

Abstract

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 P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from Mand k = 1, 2; if an operator T from P1 admits the bounded inverse T−1 then T−1 lies in P1. If a bounded operator T lies in P1 then T is normaloid, Tn belongs to P1 and a rearrangement μt(Tn) ≥ μt(T )n for all t > 0 and natural n. If a τ-measurable operator T is hyponormal and Tn is τ-compact operator for some natural number n then T is both normal and τ-compact. If an operator T lies in P1 then T 2 belongs to P1. If M= B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators onH. If a τ-measurable operator A is q-hyponormal (1 ≥ q > 0) and |A*| ≥ μ∞(A)I then Ais normal. In particular, every τ-compact q-hyponormal (or q-cohyponormal) operator is normal. Consider a τ-measurable nilpotent operator Z ≠ 0 and numbers a, b ∈ R. Then an operator Z*Z − ZZ* + aRZ + bSZ cannot be nonpositive or nonnegative. Hence a τ-measurable hyponormal operator Z ≠ 0 cannot be nilpotent.