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.)
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