  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.
