Dirac operators on Lagrangian submanifolds

We study a natural Dirac operator on a Lagrangian submanifold of a K"ahler manifold. We first show that its square coincides with the Hodge-de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.