On skew derivations and generalized skew derivations in Banach algebras.

Let A be unital prime Banach algebra over ℝ or ℂ with centre and G1, G2 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 ∈ G1, b ∈ G2, there exists an integer m ∈ Z>1 depending on a and b such that either. For each a ∈ G1, b ∈ G2, there exists an integer m ∈ 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 affirmative answer to some questions posed in [21]. [ABSTRACT FROM AUTHOR]