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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 $-\sqrt{-1}(d+\delta)\widehat{c}(V)$ on3,4-dimensional oriented compact manifolds with (resp.without) boundary.

The operator $\sqrt{-1}\widehat{c}(V)(d+\delta)$ and the Kastler-Kalau-Walze type theorems