Question about linear operators

Question about linear operators

Let $X$ be an arbitrary vector space over the reals or the complex
numbers, and let $L$ be a linear operator on $X$ such that $L^2=I$, where
$I$ is the identity operator on $X$. Must it be true that $\ker(I+L)$ is a
subset of $\operatorname{im}(I-L)$?