Spectral Properties of $k$-quasi-$*$-$A(n)$ Operators

Abstract

In this paper, we prove the following assertions: (1) If $T$ is a $k$-quasi-$*$-$A(n)$ operator, then $T$ is polaroid. (2) If $T$ is a $k$-quasi-$*$-$A(n)$ operator, then the spectrum $\sigma$ is continuous. (3) If $T$ or $T^{*}$ is a $k$-quasi-$*$-$A(n)$ operator, then Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$. (4) If $T^{*}$ is a $k$-quasi-$*$-$A(n)$ operator, then generalized $a$-Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$. Finally, the finiteness of a quasi-$*$-$A(n)$ operator is also studied.