Continuous linear maps on reflexive algebras behaving like Jordan left derivations at idempotent-product elements

Abstract: Let $\A$ be a Banach algebra with unity $\textbf{1}$ and $ \M $ be a unital Banach left $ \A $-module. let $ \delta: \A \rightarrow \M$ be a continuous linear map with the property that [ a,b\in \A, \quad ab+ba=z \Rightarrow 2a\delta(b)+2b\delta(a)=\delta(z), ] where $z\in \A$. In this article, first we characterize $\delta$ for $z=\textbf{1}$. Then we consider the case $\A=\M=Alg \mathcal{L}$, where $Alg \mathcal{L}$ is areflexive algebra on a Hilbert space $ \Hh $ and $z=P$ is a non-triavial idempotent in $\A$ with $P(\Hh) \in \mathcal{L}$ and describe $\delta$. Finally we apply the main results to $CSL$-algebras, irreducible $CDC$ algebras and nest algebras on a Hilbert space $\Hh$.