I am trying to read the heat kernel proof of the Atiyah-Singer index theorem by the method of Getzler's rescaling method. Here, we define an operator $$(\delta_{\lambda}\phi)(t,x)=\sum_{j=0}^n \lambda^{-j}\phi(\lambda^2t,\lambda x)$$ $\phi$ is defined on $\mathbb R^{+}\times U$ with values in $Cl(T_qM)\otimes End(W)$ where $W$ is the twisted bundle for Clifford module bundle $E=S\otimes W$. We can imagine $\phi(t,x)$ as terms like asymptotic expansion of heat kernels $k(t,x)$ and identify it with CLifford algebra basis.
And we define the Getzler's operator as $\delta_{\lambda} A \delta_{\lambda}^{-1}$ for some operators (For the square of the Dirac operator $D^2$, this is actually the rescaled Dirac laplacian $D_{\lambda}^2$, which has the rescaled heat kernel, this transforms the asymptotic expansion with $t\to0^{+}$ to rescaled $\lambda\to 0^{+}$)
My question is how does the inverse $\delta_{\lambda}^{-1} $ work ? How to calculate it on operators like $A=\partial_t$, the example shows that this is equal to $\delta_{\lambda} \partial_{t} \delta_{\lambda}^{-1}=\lambda^{-2}\partial_{t}$