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

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