Abstract: Let H be an infinite dimensional complex separable Hilbert space and B(H) be the algebra of all bounded linear operators on H. T \in B(H) is said to satisfy property (R_1) if \sigma _a(T)\backslash \sigma _ab(T) \subseteq \pi _00(T), where \sigma _a(T) and \sigma _ab(T) denote the approximate point spectrum and the Browder essential approximate point spectrum of T respectively, and \pi _00(T) = \ \lambda \in \rmiso\sigma (T):0 < \rmdimN(T - \lambda I) < \infty \ . If \sigma _a(T)\backslash \sigma _ab(T) = \pi _00(T), T is said to satisfy property (R). In this paper, by using the new spectrum, the necessary and sufficient conditions for which the property (R_1) or property (R) holds for bounded linear operators and its functions are given. Also, the new judgements for a-Weyl's theorem and property (R) are obtained.