On quasi-class A contractions

Let QA denote the class of bounded linear Hilbert space operators T which satisfy the operator inequality . It is proved that if T∈QA is a contraction, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator D=T∗(|T2|-|T|2)T is strongly stable. It is shown that if T∈QA is a contraction with Hilbert–Schmidt defect operator such that T-1(0)⊆T∗-1(0), then T is completely non–normal if and only if T∈C10, and a commutativity theorem is proved for contractions T∈QA. Let Tu and Tc denote the unitary part and the cnu part of a contraction T, respectively. We prove that if A=Au⊕Ac and B=Bu⊕Bc are QA-contractions such that μAc<∞, then A and B are quasi-similar if and only Au and Bu are unitarily equivalent and Ac and Bc are quasi-similar.