  In this paper we introduce the notion of property $(BR)$ and property $(BgR)$
for bounded linear operators defined on an infinite-dimensional Banach space.
These properties in connection with Weyl type theorems and in the frame of
polaroid operators are investigated. Moreover, we study the stability of these
properties under perturbations by commuting finite-dimensional,
quasi-nilpotent, Riesz and algebraic operators.
