Given $A$ and $B$, two $n \times n$ complex Hermitian positive semidefinite matrices such that $A< B$. I want to show existence (or non existence) of a real symmetric positive matrix $X$ such that \begin{equation} 0\le A\le X \le B. \end{equation} What is the most constrictive way to generate examples of $X$?
My Approach: It is clear that $X$ lies between real parts of $A$ and $B$ respectively. So natural choice is to take some convex combination of these real parts. Problem is, the resulting matrix may not satisfy the above inequality involving complex matrices $A$ and $B$. Further, I can cook up examples in $2 \times 2$ case, where such interval may not contain any real matrix, but contains complex matrices.
So the question once again: Given $A$ and $B$ complex Hermitian positive semi-definite, what is the algorithm which guarantees an output positive real matrix $X$ satisfying above inequality.