 Solution

Hadamard well-posedness requires existence, uniqueness, and continuous dependence of $u$ on $f$. A compact operator with infinite-dimensional range cannot have closed range: otherwise its inverse on the orthogonal complement of its kernel would be bounded, making the identity on an infinite-dimensional space compact. Hence the inverse on $\mathrm{ran}A$ is unbounded. If $\ker A\ne \{0\}$ uniqueness also fails, and data outside the range have no exact solution. In every case at least stability fails, so the inverse problem is ill posed.