Slope semistability of rank 2 Lazarsfeld-Mukai bundles on K3 surfaces and ACM line bundles

Previously, many people have studied a stability of vector bundles of given rank and Chern classes on algebraic varieties. Recently, we are interested in the slope stability of the rank 2 Lazarsfeld-Mukai bundle $E_{C,Z}$ on a K3 surface $X$ associated to a very ample smooth curve $C$ on $X$ and a base point free pencil $Z$ on $C$ with respect to ${\mathcal{O}}_X(C)$. In this paper, we will give a sufficient condition for such a Lazarsfeld-Mukai bundle $E_{C,Z}$ to be ${\mathcal{O}}_X(C)$-slope semistable by ACM line bundles with respect to ${\mathcal{O}}_X(C)$.