B-Fredholm and Drazin invertible operators through localized SVEP

Let $X$ a Banach space and $T$ a bounded linear operator on $X.$ We denote by $S(T)$ the set of all $\lambda \in \cit$ such that $T$ does not have the single-valued extension property at $\lambda$. In this note we prove equality up to $S(T)$ between the left Drazin spectrum and the left B-Fredholm spectrum and between the semi-essential approximate point spectrum and the left Drazin spectrum. As applications we investigate generalized Weyl's theorem for operator matrices and multipliers operators.