Abstract. For a bounded linear operator $T$ acting on a Banach space let $\sigma_{SBF_{+}^-}(T)$ be the set of all $\lambda\in {\msbm C}$ such that $T-\lambda I$ is upper semi-$B$-Fredholm and ${\rm ind} ({T- \lambda I}) \leq0$, and let $ E^a(T)$ be the set of all isolated eigenvalues of $T$ in the approximate point spectrum $ \sigma_{a}(T)$ of $T$. We say that $T$ satisfies generalized $a$-Weyl's theorem if $\sigma_{SBF_{+}^-}(T)= \sigma_{a}(T) \setminus E^a(T)$. Among other things, we show in this paper that if $T$ satisfies generalized $a$-Weyl's theorem, then it also satisfies generalized Weyl's theorem $\sigma_{BW}(T) = \sigma(T) \setminus E(T)$, where $ \sigma_{BW}(T)$ is the $B$-Weyl spectrum of $T$ and $E(T)$ is the set of all eigenvalues of $T$ which are isolated in the spectrum of $T$.

AMS Subject Classification (1991): 47A53, 47A55

Keyword(s): semi-Fredholm operator, quasi-Fredholm operator, B, semi--Fredholm operator, Weyl's theorem, a, -Weyl's theorem, a, generalized-Weyl's theorem

Received July 27, 2001, and in revised form December 12, 2001. (Registered under 2901/2009.)