Index of B-Fredholm operators and generalization of a Weyl theorem

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On the Equivalence of Weyl Theorem and Generalized Weyl Theorem (2007, background): “In the case of a normal operator T acting on a Hilbert space H, Berkani [ 3, Theorem 4.5 ] showed that σBW (T )= σ(T )\E(T ), where σBW (T )i s theB-Weyl spectrum of T and E(T ) is the set of all eigenvalues of T which are isolated in the spectrum of T . This result gives a generalization of the cla”
Weyl Type Theorems for Left and Right Polaroid Operators (2010, background): “A]. Every B-Fredholm operator T admits on a Banach space the representation T = T0 ⊕ T1 ,w hereT0 is nilpotent and T1 is Fredholm [ 18 ], so every Drazin invertible operator is B-Fredholm.”
On Drazin invertibility (2008, background): “Every B -Fredholm operator T admits the representation T = T 0 ⊕ T 1 , where T 0 is Fredholm and T 1 is nilpotent [11], so every Drazin invertible operator is B -Fredholm.”