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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?

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    $\begingroup$ What's Choi's theorem? $\endgroup$
    – Felix Goldberg
    Commented Jan 3, 2013 at 14:45
  • $\begingroup$ @FelixGoldberg en.wikipedia.org/wiki/… In short, it says that all completely positive linear maps (a stronger condition than preserving positive semidefiniteness) are precisely those in the form $\Phi(X)=\sum W_i X W_i^*$ $\endgroup$
    – Federico Poloni
    Commented Jan 3, 2013 at 15:19

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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.

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