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.