Characterizations of left derivable maps at non-trivial idempotents on nest algebras

Let $Alg\mathcal{N}$ be a nest algebra associated with the nest $\mathcal{N}$ on a (real or complex) Banach space $\text{X}$. Suppose that there exists a non-trivial idempotent $P\in Alg\mathcal{N}$ with range $P(\X) \in $\mathcal{N}$$and$\delta:Alg$\mathcal{N}$ \rightarrow Alg$\mathcal{N}$$ is a continuous linear mapping (generalized) left derivable at $P$, i.e. $\delta (ab)=a\delta (b)+b\delta (a)$ ($\delta (ab)=a\delta (b)+b\delta (a)-ba\delta (I)$) for any $a,b\in Alg$\mathcal{N}$$with$ab=P$\mathrm{.}weshowthat$\delta$ is a (generalized) Jordan left derivation. Moreover, we characterize the strongly operator topology continuous linear maps $\delta$ on some nest algebra $Alg\mathcal{N}$ with property that $\delta (P)=2P\delta (P)$ or $\delta (P)=2P\delta (P)-P\delta (I)$ every idempotent $P$ in $Alg\mathcal{N}$.