A new proof of Singer-Wermer theorem with some results on {g, h}-derivations

Singer and wermer proved that if a is a commutative banach algebra and d : a→a is a continuous derivation, then d(a)⊆rad(a), where rad(a) denotes the jacobson radical of a. in this article, we will establish a new proof of that. moreover, we prove that every continuous jordan derivation on a finite dimensional banach algebra is identically zero under certain conditions. as another objective of this article, we study {g, h}-derivations on algebras. in this regard, we prove that if f is a {g, h}-derivations on a unital algebra, then f, g and h are generalized derivations. in addition, we achieve some results concerning the automatic continuity of {g, h}-derivations on banach algebras. in the last section of this article, we introduce the concept of a {g, h}-homomorphism and then we characterize it under certain conditions. indeed, we prove that if a is an algebra with the identity element e and f: a→a is a {g, h}-homomorphism such that g(e) and h(e) are invertible elements of a, then there exists a homomorphism θ of a such that f(ab) = f(a)θ(b) = θ(a)f(b), g(ab) = g(a)θ(b) = θ(a)g(b) and h(ab) = h(a)θ(b) = θ(a)h(b) for all a, b ∈a.