Multiplicative Lie-type derivations on standard operator algebras

Let U be a standard operator algebra on a complex Banach space X, dim X > 1, and p(n)(T-1, T-2, ... , T-n) the (n - 1)th-commutator of elements T-1, T-2, ... , T-n is an element of U. Then every map xi : U -> U (not necessarily linear) satisfying xi(p(n)(T-1, T-2, ... , T-n)) = Sigma(n)(i=1) p(n)(T-1, T-2, ... , Ti-1, xi(T-i), Ti+1, ... , T-n) for all T-1, T-2, ... , T-n is an element of U is of the form xi = Omega + Gamma, where Omega : U -> U is an additive derivation and Gamma : U -> CI is a map that vanishes at each (n - 1)th-commutator p(n)(T-1, T-2, ... , T-n) for all T-1, T-2, ... , T-n is an element of U. In addition, if the map xi is linear and satisfies the above relation, then there exist an operator S is an element of U and a linear map Gamma : U -> CI satisfying Gamma(p(n)(T-1, T-2, ... , T-n)) = 0 for all T-1, T-2, ... , T-n is an element of U, such that xi(T) = [T, S] + Gamma(T) for all T is an element of U.