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Let $\A$ be a Banach algebra with unity $\textbf{1}$ and $ \M $ be a unitalBanach left $ \A $-module. let $ \delta: \A \rightarrow \M$ be a continuouslinear map with the property that \[ a,b\in \A, \quad ab+ba=z \Rightarrow2a\delta(b)+2b\delta(a)=\delta(z), \] where $z\in \A$. In this article, firstwe characterize $\delta$ for $z=\textbf{1}$. Then we consider the case$\A=\M=Alg \mathcal{L}$, where $Alg \mathcal{L}$ is areflexive algebra on aHilbert 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 mainresults to $CSL$-algebras, irreducible $CDC$ algebras and nest algebras on aHilbert space $\Hh$.

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