A strictly positive operator is invertible
Are you sure it is true?
Consider $T:\ell_2\to\ell_2$ which maps $(a_n)$ to $(a_n/n)$. This is clearly self-adjoint, and positive: $$\langle Ta,a\rangle=\sum_{n\ge 1} \frac{|a_n|^2}n$$ and this is $>0$ whenever any $a_n\ne 0$.
On the other side, $\langle Tx,x\rangle$ is not bounded from below: for the sequence $x_n=1$ and $x_k=0$ if $k\ne n$, we have $\langle Tx,x\rangle=1/n$.