In this paper, we introduces the property $(aBw)$, 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 $(aBw)$ holds. Also, we prove that if $T\in \mathbf{L}\mathbf{(}\mathbb{X}\mathbf{)}$ satisfies property $(aBw)$ then $T$ satisfies property $(Bw)$. Certain conditions are explored on Hilbert space operators $T$ and S so that $T\oplus S$ obeys property $(aBw)$.