Let $$M = \left(\begin{array}{cc}{A} & {B} \\ {B^{T}} & {C}\end{array}\right)$$ be symmetric, and let $A$ be invertible. Then the Schur Complement Lemma suggests that $$C-B^{T} A^{-1} B \succeq 0 \implies M \succeq 0$$
Is there an analogous result for the following expression? $$C-B^{T} A^{-2} B \succeq 0$$ such that it can be setup as valid linear LMI constraint in some existing SDP solver.