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

On an infinite dimensional complex Hilbert space $\mathcal{H}$, we consider a standard operator algebra $\mathcal{S}$ with an identity operator $I$ that is closed with respect to adjoint operation. $P_n\left({\mathcal{X}}_1,{\mathcal{X}}_2,{\mathcal{X}}_3,\dots ,{\mathcal{X}}_n\right)$ is set of polynomials defined under indeterminates ${\mathcal{X}}_1,{\mathcal{X}}_2,\cdots ,{\mathcal{X}}_n$ by $n$ with multiplicative Lie products with set of positive integers $\mathbb{N}.$ It is shown that a map $\mathrm{\Theta }:\mathcal{S}\to \mathcal{S}$ satisfying

$$\begin{array}{r}\mathrm{\Theta }\left(P_n\left({\mathcal{D}}_1,{\mathcal{D}}_2,{\mathcal{D}}_3,\dots ,{\mathcal{D}}_n\right)\right)=\sum _{i=1}^n{P}_n\left({\mathcal{D}}_1,\dots ,{\mathcal{D}}_{i-1},\mathrm{\Theta }\left({\mathcal{D}}_i\right),{\mathcal{D}}_{i+1},\dots ,{\mathcal{D}}_n\right),\end{array}$$

for any ${\mathcal{D}}_1,{\mathcal{D}}_2,{\mathcal{D}}_3,\dots ,{\mathcal{D}}_n\in \mathcal{S}$ with ${\mathcal{D}}_1{\mathcal{D}}_2{\mathcal{D}}_3\dots {\mathcal{D}}_n=0$ can be represented as $d(x)+\tau (x)$ for every $x\in \mathcal{S}$, where $d:\mathcal{S}\to \mathcal{S}$ is an additive derivation with another map $\tau :\mathcal{S}\to \mathcal{Z}(\mathcal{S})$ that vanishes on each $(n-1{)}^{th}$ commutator $P_n\left({\mathcal{D}}_1,{\mathcal{D}}_2,{\mathcal{D}}_3,\dots ,{\mathcal{D}}_n\right)$ with ${\mathcal{D}}_1{\mathcal{D}}_2{\mathcal{D}}_3\cdots {\mathcal{D}}_n=$ 0.