Computing the inverse of the operator $L:= f - \Delta f$.

Question

Here is an argument I read for computing the inverse of the differential operator $L:S(\Bbb R^n) \rightarrow S(\Bbb R^n)$ on Schwartz spaces where $$Lf:= f - \Delta f = f- \sum_{i=1}^n (D^i)^2 f$$

By the basic rules for the Fourier transform, $$ \widehat{Lf}(\xi) = (1+|\xi|^2) \hat{f}(\xi)$$ Therefore, $L$ is invertible with inverse $M$ given by $$ \widehat{Mf} (\xi):= (1+|\xi|^2)^{-1}\hat{f}$$

I get how we got the formula $\widehat{Lf}$ and why $M$ exists - using Fourier inversion.

But how is $M$ the inverse of $L$?

Answer 1

$\widehat {MLf} (\xi) =(1+|\xi|^{2})^{-1}) \widehat {Lf} (\xi)=(1+|\xi|^{2})^{-1})(1+|\xi|^{2})\hat {f} (\xi)=\hat {f} (\xi)$ So ML=I. Similarly, $LM=I$.