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 P_k, then UTU* belongs to P_k 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.