On operators satisfying T∗∣T2∣T ⩾ T∗∣T∣2T

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce the class, denoted QA, of operators satisfying T∗∣T2∣T ⩾ T∗∣T∣2T and we prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a non-zero isolated point λ∘ of the spectrum of T∈QA, then E is self-adjoint, and we give a necessary and sufficient condition for T ⊗ S to be in QA when T and S are both non-zero operators.