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}, \ldots, \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 $ \Theta: \mathcal{S} \rightarrow \mathcal{S} $ satisfying

$ \begin{eqnarray*} \Theta\left(P_{n}\left(\mathcal{D}_{1}, \mathcal{D}_{2}, \mathcal{D}_{3}, \ldots, \mathcal{D}_{n}\right)\right) = \sum\limits_{i = 1}^{n} P_{n}\left(\mathcal{D}_{1}, \ldots, \mathcal{D}_{i-1}, \Theta\left(\mathcal{D}_{i}\right), \mathcal{D}_{i+1}, \ldots, \mathcal{D}_{n}\right), \end{eqnarray*} $

for any $ \mathcal{D}_{1}, \mathcal{D}_{2}, \mathcal{D}_{3}, \ldots, \mathcal{D}_{n} \in \mathcal{S} $ with $ \mathcal{D}_{1} \mathcal{D}_{2} \mathcal{D}_{3}\ldots \mathcal{D}_{n} = 0 $ can be represented as $ d(x)+\tau(x) $ for every $ x \in \mathcal{S} $, where $ d: \mathcal{S} \rightarrow \mathcal{S} $ is an additive derivation with another map $ \tau: \mathcal{S} \rightarrow \mathcal{Z}(\mathcal{S}) $ that vanishes on each $ (n-1)^{th} $ commutator $ P_{n}\left(\mathcal{D}_{1}, \mathcal{D}_{2}, \mathcal{D}_{3}, \ldots, \mathcal{D}_{n}\right) $ with $ \mathcal{D}_{1} \mathcal{D}_{2} \mathcal{D}_{3}\cdots \mathcal{D}_{n} = $ 0.