(δ, ε)-DOUBLE DERIVATIONS ON BANACH ALGEBRAS

Let A be an algebra and let delta, epsilon : A -> A be two linear mappings. A (delta, epsilon)-double derivation is a linear mapping d : A -> A satisfying d(ab) = d(a)b + ad(b) + delta(a)epsilon(b)+ epsilon(a)delta(b) (a, b is an element of A). We study some algebraic properties of these mappings and give a formula for calculating d(n)(ab). We show that if A is a Banach algebra such that either is semi-simple or every derivation from A into any Banach A-bimodule is continuous then every (delta, epsilon)-double derivation on A is continuous whenever so are delta and epsilon. We also discuss the continuity of epsilon when d and delta are assumed to be continuous.