(\(\delta , \epsilon \))-double derivations on Banach algebras (Q533314)

Let \(\delta \) and \(\epsilon : \mathcal {A}\to \mathcal {A}\) be two linear mappings of algebra \(\mathcal {A}\) into itself. A linear mapping \(d: \mathcal {A}\to \mathcal {A}\) is called a (\(\delta , \epsilon \))-double derivation if \[ d(ab)=d(a)b+ad(b)+\delta (a)\epsilon (b)+\epsilon (a)\delta(b) \] holds true for all \(a, b\in \mathcal {A}\). In this paper, the authors study some algebraic properties of (\(\delta , \epsilon \))-double derivations and give a formula for calculating \(d^n(ab)\). They show that, if \(\mathcal {A}\) is a Banach algebra such that either \(\mathcal {A}\) is semi-simple or every derivation from \(\mathcal {A}\) into any Banach \(\mathcal {A}\)-module is continuous, then every (\(\delta , \epsilon \))-double derivation on \(\mathcal {A}\) is continuous whenever so are \(\delta \) and \(\epsilon \). The authors also discuss the continuity of \(\epsilon \) when \(d\) and \(\delta \) are continuous.