Characterizations of automorphisms of operator algebras on Banach spaces

Let X   be a complex Banach space of dimension greater than one, and denote by B(X)B(X) the algebra of all the bounded linear operators on X  . It is shown that if ϕ:B(X)→B(X)ϕ:B(X)→B(X) is a multiplicative map (not assumed linear) and if ϕ   is sufficiently close to a linear automorphism of B(X)B(X) in some uniform sense, then it is actually an automorphism.