# Positive square root on a real Hilbert space?

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?

• Similar question: mathoverflow.net/q/254385 – J.R. Nov 11 '16 at 9:22
• 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. – Disintegrating By Parts Nov 11 '16 at 19:28
• See my edit to see if they help at all. – Mathmank Nov 11 '16 at 20:27