$(\delta‎, ‎\varepsilon)$-double derivations on Banach algebras

Abstract

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