Characterizations of generalized derivations associated with hochschild 2-cocycles and higher derivations

Let be a unital Banach algebra and be a unital -bimodule. A bilinear mapping α : is called a Hochschild 2-cocycle if xα(y, z) ? α(xy, z) + α(x, yz) ? α(x, y)z = 0 for any . We show that if δ is a linear mapping from into satisfying δ(xy) = δ(x)y + xδ(y) + α(x, y) for any with xy = W, where is a left or right separating point of , then δ is a generalized Jordan derivation associated with a Hochschild 2-cocycle α. We also find the relation of higher derivations and generalized derivations associated with Hochschild 2-cocycles.