Additive preservers of Drazin invertible operators with bounded index

Ascent, descent and additive preserving problems

“…Hence, we get by [17,Proposition 2.17] that Ψ(T ) = T . This shows that Φ(T ) = cAT A −1 for all T ∈ B(X).…”

“…Similar results are given for F + n (X), F − n (X) and F ± n (X). It should be noted that the ascent and the hyper-kernel of an operator T ∈ B(X) are related by the following inequality (see [17])…”

“…For T, F ∈ B(X), we denote respectively byT andF the operators induced by T and F on X/M(T, F ). Note that the hyper-kernels ofT + cF and T + cF are related by the following relation (see [17,Lemma 2.9])…”

“…Throughout the sequel, Λ will denote any of the subsets B + n (X), B − n (X) or B ± n (X). Also, the subset B n (X) = B + n (X) ∩ B − n (X), introduced and studied in [17], will be used in the rest of this paper.…”

“…Assume now that F is finite-rank, and let p = min{dim ran(F ), 2}. It follows from [17,Proposition 2.12] that there exists an invertible operator T ∈ B(X) such that T + F / ∈ B n (X) and T − (−1) p F / ∈ B n (X). But, T + F and T − (−1) p F are Fredholm operators of index zero, then T +F / ∈ Λ and T −(−1) p F / ∈ Λ.…”