Let $X$ be a smooth projective surface and $C$ be a smooth curve on $X$ ( let $j$ denote the embedding of $C$ inside $X$). Let $L$ be a line bundle on $C$. Assume that it's known that $h^0( j_*(L) \otimes \mathcal O_X(-1))=0$, then can we say that $h^0( j_*(L) \otimes \mathcal O_X(-n))=0$ for all positive integer $n$?
Can we say that $j_*(L) \otimes \mathcal O_X(-n) \hookrightarrow j_*(L) \otimes \mathcal O_X(-1)$? This is true if $j_*(L)$ is a bundle. Does it also hold in other situations?
Edit (from OP's comment): Can we argue like this: One can note that, $H^0(X, j_*(L) \otimes \mathcal O_X(-m)) \cong H^0(C, L \otimes j^*(\mathcal O_X(-m))$ for any $m$ using Projection formula.
From the exactness of Pullback functor, one has $\mathcal O_X(-n) \hookrightarrow \mathcal O_X(-1)$ on $X$ induces $j^*(\mathcal O_X(-n) )\hookrightarrow j^*(\mathcal O_X(-1))$ on $C$. Tensoring this with the line bundle $L$ and taking cohomolgy one has $$h^0(L \otimes j^*(\mathcal O_X(-n) ) \leq h^0(L \otimes j^*(\mathcal O_X(-1) ) =0$$ and hence we have the desired result. Please correct me if this is wrong.