TITLE: Linear maps preserving positive semidefiniteness
QUESTION [5 upvotes]: I know of Choi's theorem and some related problems, but not a solution to this exact problem:

Characterize the linear maps from the space $S_n$ of symmetric $n \times n $ matrices to itself that preserve positive semidefiniteness.

It looks a natural question; has a simple characterization been found? Where can I find it?

REPLY [4 votes]: Somewhat surprisingly, this seems to be still open. It is a linear preserver problem, about which there is a nice overview here. But your specific problem seems to be open, according to this recent preprint. They also say that in an earlier paper they settled the problem with some extra assumptions.