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

Abstract: 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.