Essentially Invertible Measurable Operators Affiliated to a Semifinite von Neumann Algebra and Commutators

Suppose that a von Neumann operator algebra M acts on a Hilbert space H and τ is a faithful normal semifinite trace on M . If Hermitian operators X,Y∈ S(M,τ) are such that -X≤ Y≤ X and Y is τ -essentially invertible then so is X . Let 0<p≤ 1 . If a p -hyponormal operator A∈ S(M,τ) is right τ -essentially invertible then A is τ -essentially invertible. If a p -hyponormal operator A∈B(H) is right invertible then A is invertible in B(H) . If a hyponormal operator A∈ S(M,τ) has a right inverse in S(M,τ) then A is invertible in S(M,τ) . If A,T∈M and μₜ(Aⁿ)1/n→ 0 as n→∞ for every $ t>0 $ then $ AT $ ( $ TA $ ) has no right (left) τ -essential inverse in S(M,τ) . Suppose that H is separable and H=∞ . A right (left) essentially invertible operator A∈B(H) is a commutator if and only if the right (left) essential inverse of A is a commutator.