  In this paper, we obtain two Lichnerowicz type formulas for the operators
$\sqrt{-1}\widehat{c}(V)(d+\delta)$ and $-\sqrt{-1}(d+\delta)\widehat{c}(V)$.
And we give the proof of Kastler-Kalau-Walze type theorems for the operators
$\sqrt{-1}\widehat{c}(V)(d+\delta)$ and $-\sqrt{-1}(d+\delta)\widehat{c}(V)$ on
3,4-dimensional oriented compact manifolds with (resp.without) boundary.
