MODULE GENERALIZED DERIVATIONS ON TRIANGULAUR BANACH ALGEBRAS

Let $A_1$, $A_2$ be unital Banach algebras and $X$ be an $A_1$-$A_2$- module. Applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $A_i$ into the dual space $A^*_i$ (for$i=1,2$) and such derivations  from  the triangular Banach algebraof the form $mathcal{T} :=left(begin{array}{lc} A_1 &X 0  & A_2end{array}right)$  into the associated triangular $mathcal{T}$-  bimodule $mathcal{T}^*$ of theform $mathcal{T}^*:=left(begin{array}{lc} A_1^* &X^* 0  & A_2^*end{array}right)$. In particular, we show that the  so-called generalized first cohomology group from $mathcal{T}$ to $mathcal{T}^*$ is isomorphic to the directed sum of the generalized  first  cohomology group from $A_1$ to $A^*_1$ and the generalized  first cohomology group from $A_2$ to $A_2^*$