# Positive unbounded operators

Let $T$ be an operator in $H$. We say self adjoint $T$ is positive iff $(\forall x\in H)\langle Tx,x\rangle \geq 0$. As in the case of bounded operators, it is true that a self-adjoint operator $T$ is positive iff its spectrum $\sigma(T)\subset[0,\infty)$. The 'only if' part is proved in a similar way as for bounded operators. The 'if' part is proved by using the integral representation for self adjoint unbounded operators. Is there a way to prove it without using this spectral theory?

Similarly, the fact that every positive (unbounded) operator has a positive square root is proved using the spectral theorem. Is there a way to prove this without using it?

• I think continuous function calculus is a bit easier to construct than the general spectral theorem, though I'm not sure if it's still true for unbounded operators. What's wrong with spectral theorem anyway? – tomasz Jan 21 '14 at 2:14
• The proof for the spectral theorem for unbounded self-adjoint operators requires a measurable(not necessarily bounded measurable) functional calculus. Of course once that is proved the above two results follow as easy corollaries. I am trying to see if they can be proved independently. – Arundhathi Jan 21 '14 at 2:26
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