Suppose that $C$ and $D$ are (real) symmetric positive definite matrices and that $A$ is a matrix such that
$$ H = (C + A^T D A)^{-1}A^TD^2 A(C + A^T D A)^{-1} $$
is well defined. Is the maximum eigenvalue of $H$ bounded as a function of $D$? For scalars it clearly holds, if $A\neq 0$, that $H = A^2D^2 / (C + A^2D)^2 \to A^{-2}$ as $D \to \infty$ and $H \to 0$ as $D \to 0$.