B-Weyl spectrum and poles of the resolvent

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Let T be a bounded linear operator acting on a Banach space and let σ BW (T)={λ∈ C such that T−λI is not a B-Fredholm operator of index 0} be the B-Weyl spectrum of T. Define also E( T) to be the set of all isolated eigenvalues in the spectrum σ( T) of T, and Π( T) to be the set of the poles of the resolvent of T. In this paper two new generalized versions of the classical Weyl's theorem are considered. More precisely, we seek for conditions under which an operator T satisfies the generalized Weyl's theorem: σ BW( T)= σ( T)⧹ E( T), or the version II of the generalized Weyl's theorem: σ BW( T)= σ( T)⧹ Π( T).