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

Abstract

Let $n$ be a positive integer, and an operator $T\in B(\mathscr{H})$ is called a class $A(n)$ operator if ${\left|T^{1+n}\right|}^{2/\left(1+n\right)}\ge {|T|}^2$ and $n$-paranormal operator if ${∥T^{1+n}x∥}^{1/\left(1+n\right)}\ge ||Tx||$ for every unit vector $x\in \mathscr{H}$, 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 $T\otimes S$ 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.