Assume $A$ is a $m \times n$ ($m>n$) semi-orthogonal matrix such that $A^TA=I$, and $D$ is a diagonal matrix with positive diagonal elements. Let us define
$$T(a) = (A^TDPA)(A^TPDPA)^{-1}(A^TDPA)$$
where $P = (D+aI)^{-1}$ and $a>0$. Equivalently,
$$T(a) = (A^TD(D+aI)^{-1}A)(A^T(D+aI)^{-1}D(D+aI)^{-1}A)^{-1}(A^TD(D+aI)^{-1}A)$$
$T(a)$ is a positive definite matrix and I want to show that if $a>b$ then $T(a) > T(b)$ with the meaning that $T(a)-T(b)$ is a positive definite matrix.
In other words, note $P$ is a positive definite matrix and the question can be "is $T(P)$ a monotonic decreasing function of $P$, $P>Q$ then $T(P)<T(Q)$?".
