B-Weyl spectrum and poles of the resolvent

Let T be a bounded linear operator acting on a Banach space and let σBW(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: σBW(T ) = σ(T )\E(T ), or the version II of the generalizedWeyl’s theorem: σBW(T ) = σ(T )\Π(T ).  2002 Elsevier Science (USA). All rights reserved.