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}. Does this family converge to the identity in the L^2 operator norm? Why or why not?