Weyl's theorem for algebraically quasi-$*$-$A$ operators

Abstract

In the paper, we prove the following assertions: (1) If $T$ is an algebraically quasi-$*$-$A$ operator, then $T$ is polaroid. (2) If $T$ or $T^{*}$ is an algebraically quasi-$*$-$A$ operator, then Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$. (3) If $T^{*}$ is an algebraically quasi-$*$-$A$ operator, then a-Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$.