$(m, n)$-Jordan Derivations

Abstract

A subspace lattice $\mathcal{L}$ on $H$ is called \textit{commutative subspace lattice} if all projections in $\mathcal{L}$ commute pairwise. It is denoted by $CSL$. If $\mathcal{L}$ is a $CSL$, then $alg\mathcal{L}$ is called a $CSL$ algebra. Under the assumption $m + n \neq 0$ where $m, n$ are fixed integers, if $\delta$ is a mapping from $\mathcal{L}$ into itself satisfying the condition $(m+n)\delta(A^{2}) = 2m\delta(A)A+2nA\delta(A)$ for all $A \in \mathcal{A}$, we call $\delta$ an \textit{$(m, n)$ Jordan derivation}. We show that if $\delta$ is a norm continuous linear $(m, n)$ mapping from $\mathcal{A}$ into it self
then $\delta$ is a $(m, n)$-Jordan derivation.