On continuous linear generalized derivation in rings and Banach algebras

Abstract

In this paper, we investigate the commutativity of a prime Banach algebra [Formula: see text] which admits a nonzero continuous linear generalized derivation [Formula: see text] associated with continuous linear derivation [Formula: see text] such that either [Formula: see text] or [Formula: see text] for intergers [Formula: see text] and [Formula: see text] and sufficiently many [Formula: see text]. Further, similar results are also obtained for unital prime Banach algebra [Formula: see text] which admits a nonzero continuous linear generalized derivations [Formula: see text] satisfying either [Formula: see text] or [Formula: see text] for an integer [Formula: see text] and sufficiently many [Formula: see text].