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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.

A brief note on the spectrum of the basic Dirac operator