Abstract Let π {\mathfrak{A}} be a standard operator algebra on a complex Banach space π {\mathfrak{X}} , dim β‘ π > 1 {\dim\mathfrak{X}>1} , and p n β’ ( T… Click to show full abstract
Abstract Let π {\mathfrak{A}} be a standard operator algebra on a complex Banach space π {\mathfrak{X}} , dim β‘ π > 1 {\dim\mathfrak{X}>1} , and p n β’ ( T 1 , T 2 , β¦ , T n ) {p_{n}(T_{1},T_{2},\dots,T_{n})} the ( n - 1 ) {(n-1)} th-commutator of elements T 1 , T 2 , β¦ , T n β π {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} . Then every map ΞΎ : π β π {\xi:\mathfrak{A}\rightarrow\mathfrak{A}} (not necessarily linear) satisfying ΞΎ β’ ( p n β’ ( T 1 , T 2 , β¦ , T n ) ) = β i = 1 n p n β’ ( T 1 , T 2 , β¦ , T i - 1 , ΞΎ β’ ( T i ) , T i + 1 , β¦ , 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 , β¦ , T n β π {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} is of the form ΞΎ = Ξ© + Ξ {\xi=\Omega+\Gamma} , where Ξ© : π β π {\Omega:\mathfrak{A}\rightarrow\mathfrak{A}} is an additive derivation and Ξ : π β β β’ I {\Gamma:\mathfrak{A}\rightarrow\mathbb{C}I} is a map that vanishes at each ( n - 1 ) {(n-1)} th-commutator p n β’ ( T 1 , T 2 , β¦ , T n ) {p_{n}(T_{1},T_{2},\dots,T_{n})} for all T 1 , T 2 , β¦ , T n β π {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} . In addition, if the map ΞΎ is linear and satisfies the above relation, then there exist an operator S β π {S\in\mathfrak{A}} and a linear map Ξ : π β β β’ I {\Gamma:\mathfrak{A}\rightarrow\mathbb{C}I} satisfying Ξ β’ ( p n β’ ( T 1 , T 2 , β¦ , T n ) ) = 0 {\Gamma(p_{n}(T_{1},T_{2},\dots,T_{n}))=0} for all T 1 , T 2 , β¦ , T n β π {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} , such that ΞΎ β’ ( T ) = [ T , S ] + Ξ β’ ( T ) {\xi(T)=[T,S]+\Gamma(T)} for all T β π {T\in\mathfrak{A}} .
               
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