On skew derivations and generalized skew derivations in Banach algebras

Let A be unital prime Banach algebra over Double-struck capital R or DOUBLE-STRUCK CAPITAL C with centre and G(1), G(2) be open subsets of be a continuous linear generalized skew derivation, and be a continuous linear map. We prove that must be commutative if one of the following conditions holds:For each a is an element of G(1), b is an element of G(2), there exists an integer m is an element of Z(>1) depending on a and b such that either . For each a is an element of G(1), b is an element of G(2), there exists an integer m is an element of Z(>1) depending on a and b such that either . These results generalize a number of theorems of this type. In particular, as an application, we give an axfb03;rmative answer to some questions posed in [21].