On ∗-paranormal contractions and properties for ∗-class A operators

An operator T∈B(H) is called a ∗-class A operator if |T2|⩾|T∗|2, and T is said to be ∗-paranormal if ‖T∗x‖2⩽‖T2x‖ for every unit vector x in H. In this paper we show that ∗-paranormal contractions are the direct sum of a unitary and a C.0 completely non-unitary contraction. Also, we consider the tensor products of ∗-class A operators.