In this paper, we introduces the property $(a B w)$, a variant of generalized a-Weyl’s theorem for a bounded linear operator $T$ on an infinite-dimensional Banach space $\mathbb X$. We establish several sufficient and necessary conditions for which property $(a B w)$ holds. Also, we prove that if $T\in \mathbf {L(\mathbb {X})}$ satisfies property $(a B w)$ then $T$ satisfies property $(B w)$. Certain conditions are explored on Hilbert space operators $T$ and S so that $T \oplus S$ obeys property $(a B w)$.