In this paper, we study the stability of extended Weyl and Browdertype theorems for orthogonal direct sum S⊕T, where S and T are bounded linear operators acting on Banach space. Two counterexamples shows that property (ab), in general, is not preserved under direct sum. Nonetheless, and under the assumptions that Π
We introduce a new class which generalizes the class of B-Weyl operators. We say that T ∈ L(X) is pseudo B-Weyl if T = T 1 ⊕ T 2 where T 1 is a Weyl operator and T 2 is a quasi-nilpotent operator. We show that the corresponding pseudo B-Weyl spectrum σ pBW (T ) satisfies the equalityis the generalized Drazin spectrum of T ∈ L(X) and S(T ) (resp., S(T * )) is the set where T (resp., T * ) fails to have SVEP. We also investigate the generalized Drazin invertibility of upper triangular operator matrices by giving sufficient conditions which assure that the generalized Drazin spectrum or the pseudo B-Weyl spectrum of an upper triangular operator matrices is the union of its diagonal entries spectra.1991 Mathematics Subject Classification. Primary 47A53, 47A10, 47A11.
It is well known that an hyponormal operator satisfies Weyl's theorem. A result due to Conway shows that the essential spectrum of a normal operator N consists precisely of all points in its spectrum except the isolated eigenvalues of finite multiplicity, that's σe(N ) = σ(N ) \ E 0 (N ). In this paper, we define and study a new class named (We) of operators satisfying σe(T ) = σ(T ) \ E 0 (T ), as a subclass of (W ). A countrexample shows generally that an hyponormal does not belong to the class (We), and we give an additional hypothesis under which an hyponormal belongs to the class (We). We also give the generalisation class (gWe) in the contexte of B-Fredholm theory, and we characterize (Be), as a subclass of (B), in terms of localized SVEP.
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