Characterizing N-type derivations on standard operator algebras by local actions

On an infinite dimensional complex Hilbert space H, we consider a standard operator algebra S with an identity operator I that is closed with respect to adjoint operation. Pn n (X1, X 1 , X2, 2 , X3,. 3 ,. . . , X n ) is set of polynomials defined under indeterminates X1, 1 , X2, 2 , <middle dot> <middle dot> <middle dot>, , X n by n with multiplicative Lie products with set of positive integers N. . It is shown that a map Theta : S- S satisfying n X Theta (Pn P n ( D 1 , D2, 2 , D3,..., 3 ,..., D n )) = i = 1 P n ( D 1 ,. . . , D i -1 , Theta (Di) D i ) , Di+1, i + 1 , ... , Dn), n ) , for any D1, 1 , D2, 2 , D3,. 3 ,. . . , D n is an element of S with D 1 D 2 D 3 ... D n = 0 can be represented as d ( x ) + tau ( x ) for every x is an element of S, where d : S- S is an additive derivation with another map tau : S- Z(S) that vanishes on each (n n - 1)th th commutator Pn n (D1, D 1 , D2, 2 , D3,.. 3 ,.. . , D n ) with D 1 D 2 D 3 <middle dot> <middle dot> <middle dot>Dn D n = 0.