Generalized Derivations on Modules

Let $A$ be a Banach algebra and $M$ be a Banach right $A$-module. A linear map $\delta : M\to M$ is called a generalized derivation if there exists a derivation $d : A \to A$ such that $$\delta(xa)=\delta(x)a + x d(a) \quad (a \in A, x \in M).$$ In this paper, we associate a triangular Banach algebra ${\mathcal T}$ to Banach $A$-module $M$ and investigate the relation between generalized derivations on $M$ and derivations on ${\mathcal T}$. In particular, we prove that the so-called generalized first cohomology group of $M$ is isomorphic to the first cohomology group of ${\mathcal T}$.