Some remarks on derivations on the algebra of operators in Hilbert pro-C*-bimodules

Suppose $A$ is a pro-C*-algebra. Let $L_A(E)$ be the pro-C*-algebra of adjointable operators on a Hilbert $A$-module $E$ and let $K_A(E)$ be the closed two sided $\ast$-ideal of all compact operators on $E$. We prove that if $E$ be a full Hilbert $A$-module, the innerness of derivations on $K_A(E)$ implies the innerness of derivations on $L_A(E)$. We show that if $A$ is a commutative pro-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on $K_A(E)$ is zero. Moreover, if $A$ is a commutative $\sigma$-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on $L_A(E)$ is zero, too.