Let $L$ be a vector space with complex structure and $L_{\mathbb{C}}$ the $\mathbb{C}$-space obtained with this structure, let $T: L \rightarrow L$ be an $\mathbb{R}$-linear transformation in $L$.
The question is what conditions $T$ must satisfy in order to be seen as a linear transformation in $L_{\mathbb{C}}$.
I would like an idea of how to start thinking about this problem.
A complex structure on $L$ is an $\mathbb R$-linear map $J\colon L\to L$ satisfying $J^2 = -\operatorname{id}_L$ and you then define the $\mathbb C$-vector space structure on $L_{\mathbb C}$ by $$ (a+\mathrm{i} b) v = av+J(bv) $$ for $a,b\in\mathbb R$ and $\mathrm{i}\in\mathbb C$ the imaginary unit.
Given an $\mathbb R$-linear transformation $T\colon L\to L$ it does already satisfy additivity $T(v+w)=T(v)+T(w)$ for all $v,w\in L$ and $\mathbb R$-homogeneity $T(av)=a\,T(v)$ for all $v\in L$ and $a\in\mathbb R$. However, as a map $T_{\mathbb C}\colon L_{\mathbb C}\to L_{\mathbb C}$ it might not be $\mathbb C$-homogeneous. For it to be $\mathbb C$-homogeneous and hence $\mathbb C$-linear we must in addition have $$ T_{\mathbb C}(\mathrm i v)= \mathrm i\,T_{\mathbb C}(v) $$ for all $v\in L$, which translates back to $$ T(J(v)) = J(T(v)). $$
We conclude that an $\mathbb R$-linear map $T\colon L\to L$ induces a $\mathbb C$-linear map $T_{\mathbb C}\colon L_{\mathbb C}\to L_{\mathbb C}$ if and only if $$ T\circ J = J\circ T. $$