  In this paper, we consider the eigenvalue problem of Dirac operator on a
compact Riemannian manifold isometrically immersed into Euclidean space and
derive some extrinsic estimates for the sum of arbitrary consecutive $n$
eigenvalues of the square of the Dirac operator acting on some Dirac invariant
subbundles. As some applications, we deduce some eigenvalue inequalities on the
compact submanifolds immersed into Euclidean space, unit sphere or projective
spaces and further get some bounds of general Reilly type. In addition, we also
establish some universal bounds under certain curvature condition and on the
meanwhile provide an alternative proof for Anghel's result. In particular,
utilizing Atiyah-Singer index theorem, we drive an upper bound estimate for the
sum of the first $n$ nontrivial eigenvalues of Atiyah-Singer Laplacian acting
on the spin manifolds without dimensional assumption.
