Multiplicative Lie-type derivations on standard operator algebras

Let {\mathfrak{A}} be a standard operator algebra on a complex Banach space {\mathfrak{X}}, dim ⁡ > 1 {\dim\mathfrak{X}>1}, and p n ⁢ (T 1, T 2, T n) {p_{n}(T_{1},T_{2},\dots,T_{n})} the (n - 1) {(n-1)} th-commutator of elements T 1, T 2, T n {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}}. Then every map ζ: → {\xi:\mathfrak{A}\rightarrow\mathfrak{A}} (not necessarily linear) satisfying ζ ⁢ (p n ⁢ (T 1, T 2, T n)) = ∑ i = 1 n p n ⁢ (T 1, T 2, T i - 1, ζ ⁢ (T i), T i + 1, T n) {\xi(p_{n}(T_{1},T_{2},\dots,T_{n}))=\sum_{i=1}^{n}p_{n}(T_{1},T_{2},\dots,T_{% i-1},\xi(T_{i}),T_{i+1},\dots,T_{n})} for all T 1, T 2, T n {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} is of the form ζ = ω + Γ {\xi=\Omega+\Gamma}, where ω: → {\Omega:\mathfrak{A}\rightarrow\mathfrak{A}} is an additive derivation and Γ: → I {\Gamma:\mathfrak{A}\rightarrow\mathbb{C}I} is a map that vanishes at each (n - 1) {(n-1)} th-commutator p n ⁢ (T 1, T 2, T n) {p_{n}(T_{1},T_{2},\dots,T_{n})} for all T 1, T 2, T n {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}}. In addition, if the map ζ is linear and satisfies the above relation, then there exist an operator S {S\in\mathfrak{A}} and a linear map Γ: → I {\Gamma:\mathfrak{A}\rightarrow\mathbb{C}I} satisfying Γ ⁢ (p n ⁢ (T 1, T 2, T n)) = 0 {\Gamma(p_{n}(T_{1},T_{2},\dots,T_{n}))=0} for all T 1, T 2, T n {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}}, such that ζ ⁢ (T) = [ T, S ] + Γ ⁢ (T) {\xi(T)=[T,S]+\Gamma(T)} for all T {T\in\mathfrak{A}}.