Let $V$ be a vector bundle over an algebraic curve $C$, ad assume that that $V \cong V \otimes L$ for some line bundle $L$.

If $V$ is decomposable this is clearly possible, for example take $V \cong \mathcal{O}_C \oplus L$ with $L^2 \cong \mathcal{O}_C$.

Q.Does $V \cong V \otimes L$ imply $L \cong \mathcal{O}_C$ if $V$ is simple?