Commutators and hyponormal operators on a Hilbert space

Abstract

Let H be an innite-dimensional Hilbert space over the field C, B(H) be the ∗-algebra of all linear bounded operators on H. An operator A ∈ B(H) is a commutator, if A = [S, T ] = ST - T S for some S, T ∈ B(H). Let X, Y ∈ B(H) and X ≥ 0. If the operator XY is a non-commutator, then X^pY X^{1-p} is a non-commutator for every 0 ( p ( 1. Let A ∈ B(H) be p-hyponormal for some 0 ( p ≤ 1. If |A^∗|^r is a non-commutator for some r ) 0, then |A|^q is a non-commutator for every q ) 0. Let H be separable and A ∈ B(H) be a non-commutator. If A is hyponormal (or cohyponormal), then A is normal. We also present results in the case of a finite-dimensional Hilbert space.