Abstract. If $T$ or $T^{\ast }$ is an algebraically quasi-class $(A, k)$ operator acting on an infinite-dimensional separable Hilbert space, then we prove that generalized Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$, where $H(\sigma(T))$ denotes the set of all analytic functions in a neighborhood of $\sigma(T)$. Moreover, if $T^{\ast }$ is an algebraically quasi-class $(A, k)$ operator, then generalized $a$-Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$. Also, we prove that the spectrum, Weyl spectrum and Browder spectrum are continuous on the class of all quasi-class $(A, k)$ operators.

AMS Subject Classification (1991): 47A10, 47A53, 47B20

Keyword(s): algebraically quasi-class $(A, k)$ operator, generalized Weyl's theorem, generalized $a$-Weyl's theorem, continuity of the spectrum

Received September 24, 2010, and in final form January 26, 2011. (Registered under 69/2010.)