Abstract. Property $(gR)$ holds for a bounded linear operator $T\in L(X)$, defined on a complex Banach space $X$, if the isolated points of the spectrum $\sigma(T)$ of $T$ which are eigenvalues are exactly those points $\lambda $ of the approximate point spectrum such that $\lambda I-T$ is left Drazin invertible. In this paper we introduce this property and give some perturbation results.

AMS Subject Classification (1991): 47A10, 47A11; 47A53, 47A55

Keyword(s): property $(R)$, Weyl type theorems

Received February 7, 2012, and in revised form May 21, 2012. (Registered under 7/2012.)