1 4 Se p 20 08 A brief note on the spectrum of the basic Dirac operator

In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation (M,F) with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O’Neill tensor and the first eigenvalue of the Dirac operator on M . We discuss examples and also define a new version of the basic Laplacian whose spectrum does not depend on the choice of bundle-like metric.