Generalized derivable mappings and centralizable mappings on B(X)

Let X be a Banach space and B(X) be the algebra of all bounded linear operators on X. Firstly, we prove that every derivable mapping at a nonzero element in B(X) is a derivation. As a corollary, we show that every generalized derivable mapping on B(X) is a general derivation. Secondly, we prove that every centralizable mapping on B(X) is a centralizer.