Tensor products and the spectral continuity for $k$-quasi-$\ast$-class A operators

Abstract

An operator $T \in B( \mathcal{H}) $ is called $k$-quasi-$\ast$-class A if $T^{\ast k}(|T^{2}|-|T^{\ast}|^{2})T^{k} \geq 0$ for a positive integer $k$, which is a common generalization of $\ast$-class A and quasi-$\ast$-class A. In this paper, firstly we prove some inequalities of this class of operators; secondly we consider the tensor products for $k$-quasi-$\ast$-class A operators, giving a necessary and sufficient condition for $T\otimes S$ to be a $k$-quasi-$\ast$-class A operator when $T$ and $S$ are both non-zero operators; at last we prove that the spectrum is continuous on the class of all $k$-quasi-$\ast$-class A operators.