On Properties of Class A(n) and n-Paranormal Operators

Abstract

Let n be a positive integer, and an operator T∈B(ℋ) is called a class A(n) operator if T1+n2/1+n≥|T|2 and n-paranormal operator if T1+nx1/1+n≥||Tx|| for every unit vector x∈ℋ, which are common generalizations of class A and paranormal, respectively. In this paper, firstly we consider the tensor products for class A(n) operators, giving a necessary and sufficient condition for T⊗S to be a class A(n) operator when T and S are both non-zero operators; secondly we consider the properties for n-paranormal operators, showing that a n-paranormal contraction is the direct sum of a unitary and a C.0 completely non-unitary contraction.