Let $X$ be a positive semidefinite matrix. Is the mapping $X\to X^{-2}$ convex?
Update: or is $Tr[X^{-2} K]$ convex for PSD $X$ and $K$?
Let $X$ be a positive semidefinite matrix. Is the mapping $X\to X^{-2}$ convex?
Update: or is $Tr[X^{-2} K]$ convex for PSD $X$ and $K$?
This function is not operator convex. Counterexamples are trivial to find (just pick some random psd matrices, you'll see).
However, that said, it is true that the map $X \mapsto X^{-p}$ is operator convex for $0 < p < 1$, which essentially follows from operator convexity of $X^{-1}$.
EDIT: The updated question asks a much simpler question, whether $\text{tr}(X^{-2}K)$ is convex for psd $K$. I concluded there a bit hastily that using $K=\sum_i v_iv_i^T$, then using linearity of trace the question boils down to verifying convexity of $\text{tr}(X^{-2}vv^T)=v^TX^{-2}v$. But actually, this function is not convex, and again one can find easy numerical counterexamples.
An exception is the choice $K=I$, for which the claim holds.