Jordan higher all-derivable points in triangular algebras

Abstract: Let ${\mathcal{T}}$ be a triangular algebra. We say that $D={D_{n}: n\in N}\subseteq L({\mathcal{T}})$ is a Jordan higher derivable mapping at $G$ if $D_{n}(ST+TS)=\sum_{i+j=n}(D_{i}(S)D_{j}(T)+D_{i}(T)D_{j}(S))$ for any $S,T\in {\mathcal{T}}$ with $ST=G$. An element $G\in {\mathcal{T}}$ is called a Jordan higher all-derivable point of ${\mathcal{T}}$ if every Jordan higher derivable linear mapping $D={D_{n}}_{n\in N}$ at $G$ is a higher derivation. In this paper, under some mild conditions on ${\mathcal{T}}$, we prove that some elements of ${\mathcal{T}}$ are Jordan higher all-derivable points. This extends some results in [6] to the case of Jordan higher derivations.