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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?

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  • $\begingroup$ Similar question: mathoverflow.net/q/254385 $\endgroup$ – J.R. Nov 11 '16 at 9:22
  • $\begingroup$ Are you asking for the existence of a positive square root? Or do you know that a positive square root exists, and you want to show that such a thing is not unique? I'm not quite clear on the phrasing of your question. $\endgroup$ – Disintegrating By Parts Nov 11 '16 at 19:28
  • $\begingroup$ See my edit to see if they help at all. $\endgroup$ – Mathmank Nov 11 '16 at 20:27

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