Lie triple centralizers and generalized Lie triple derivations on triangular operator algebras by local actions

Abstract

Let $\mathcal{U}$ be a triangular operator algebra, and $\phi: \mathcal{U}\rightarrow \mathcal{U}$ be a linear map. In this paper, under some mild conditions on $\mathcal{U}$, we prove that if $\phi$ satisfies
$$
\phi([[U, V], W])=[[\phi(U), V], W]=[[U, \phi(V)], W]
$$
for any $U, V, W\in \mathcal{U}$ with $UV=UW=P$ the standard idempotent (resp. $UV=UW=0$), then there exist $\lambda\in \mathcal{Z}(\mathcal{U})$ and a linear map $\tau:\mathcal{U} \rightarrow \mathcal{Z}(\mathcal{U}) $ satisfying $\tau([[U, V], W])=0$ for any $U, V, W\in \mathcal{U}$ with $UV=UW=P$ (resp. $UV=UW=0$) such that $\phi(U)=\lambda U+\tau(U)$ for $U\in \mathcal{U}$. As an application, we give a characterization of generalized Lie triple derivations on $\mathcal{U}$.