B-Weyl spectrum and poles of the resolvent

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On the Property (gw) (2008, background): “We say that generalized Weyl’s theorem holds for T if σ(T ) \ σBW (T )= E(T ); where E(T ) is the set of all isolated eigenvalues of T, and generalized Browder’s theorem holds for T if σ(T ) \ σBW (T )=Π (T ), where Π(T ) is the set of all poles of T. Generalized Weyl’s theorem has been studied in [”
New extended Weyl type theorems (2010, unknown): “Following [6], we say that generalized Weyl’s theorem holds for T ∈ L(X) if σ(T ) \ σBW (T ) = E(T ), where E(T ) = {λ ∈ isoσ(T ) : 0 < α(T − λI)} is the set of all isolated eigenvalues of T , and that generalized Browder’s theorem holds for T ∈ L(X) if σ(T ) \ σBW (T ) = Π(T ).”