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?