I've found several references stating that when $T: H \to H$ is a positive, bounded linear operator on a complex Hilbert space, it has a unique positive square root, i.e. a positive, bounded linear operator $S$ such that $S^2=T$.
What can we say when $H$ is a real Hilbert space? Does there always exist a positive square root, but which is non-unique? What kind of statements about the existence and uniqueness of a positive square root hold in this case?