Restriction of an operator to the range of its powers

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Index of B-Fredholm operators and generalization of a Weyl theorem (2002, result): “This paper is a continuation of our previous works [2], [3], [4], [5].”
On Drazin invertibility (2008, background): “By Proposition 3.2 of [10] if T quasi-Fredholm, in particular if T is semi-B -Fredholm, then there exists n ∈ N such that T n ( X ) is closed and T [ n ] is semi-regular (i.e., it has closed range and its kernel is contained in the range of each iterate of T [ n ] ).”