Jordan higher all-derivable points in triangular algebras

TL;DR

This paper investigates the conditions under which elements in triangular algebras are Jordan higher all-derivable points, extending previous results to the context of Jordan higher derivations.

Contribution

It establishes that certain elements in triangular algebras are Jordan higher all-derivable points, broadening the understanding of derivation properties in algebraic structures.

Findings

Identifies conditions making elements Jordan higher all-derivable points

Extends previous results to Jordan higher derivations

Provides new insights into derivation structures 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.