The Dirac-Ricci operator

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Let $(M,g)$ be a spin manifold with Ricci curvature $Ric$ viewed as an endomorphism of the tangent bundle. The Dirac-Ricci operator $DR$ acting on spinors may be defined as:

$$DR(\psi)=\sum_i Ric(e_i).\nabla_{e_i}(\psi)$$

with $\psi$ a spinor and $(e_i)$ is an orthonormal basis of the tangent bundle.

The Dirac-Lichnerowicz-Ricci formula is:

$$DR^2 = -\Delta_{R} +\alpha$$

with:

$$\Delta_{R}= \sum_{i,j} g(Ric(e_i),Ric(e_j))[\nabla_{e_i}\nabla_{e_j}-\nabla_{\nabla_{e_i}e_j}]$$

$\alpha$ is a lower term; it is a scalar if $\nabla Ric=0$. If $M$ is an Einstein manifold, then the Dirac-Ricci operator is reduced to the Dirac operator.  Can we have bounds over the first proper value of the Dirac-Ricci operator?

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