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 $a_2^*$