Generalized derivations preserving quasinilpotent elements in Banach algebras

Let A be a complex Banach algebra, let Q_A be the set of all quasinilpotent elements in A and let Q_A^r be the set of all quasi-regular elements in A. We characterize generalized derivations g of A such that g(Q_A) ⊆ Q_A^r provided that A has the property β. The class of Banach algebras with the property β is quite large: it includes C*-algebras, group algebras of locally compact groups, commutative Banach algebras, Banach algebras of all bounded linear operators on Banach spaces and so on. Our theorems are natural generalizations of the recent results for derivations obtained by Alaminos et al. [Bull. London Math. Soc. 46:379–384, 2014].