@fedja I see. I think it's a great example. Here $arcsin(X)$ is positive definite but $X$ is only positive semi-definite. We can find an invertible matrix $P$ such that $arcsin(X) = P^{T} P$ and $X = P^{T} D P$, with $D$ being a diagonal matrix with some diagonal entry being zero. Thus, $aX - arcsin(X) = P^{T} (aD - I) P$ can not be positive semi-definite.

@fedja Nice example! Thanks so much!