Let $A\_t$ be family of second order, positive, elliptic differential operator mapping Sobolev $H^2$ of a compact smooth manifold (or bounded domain) to L^2. Suppose that the coefficients of $A\_t$ converge uniformly in $C^k$ for every $k$ to the coefficients of a second order, positive, elliptic differential operator $A$. $A$ is invertible (with domain L^2 and range H^2) and so we may consider the sequence $A\_t \circ A\_0^{-1}$ of operators from $L^2$ to $L^2$. Does this family converge to the identity in the $L^2$ operator norm? Why or why not?



Convergence of elliptic operators
It suffices to show that the $L\_2$ operator norm of $A\_t\circ A\_0^{-1} - I = (A\_t - A\_0)\circ A\_0^{-1}$ is small if $t$ is sufficiently small. To do this, it suffices to show that the operator norm of $A\_t - A\_0$, as map from $H^2$ to $L\_2$ is small if $t$ is small. But a linear second order operator like this has small operator norm, if the $C^0$ norm of the coefficients are small. So the fact that the coefficients converge in the $C^0$ norm gives what you want.


