Suppose that $H$ is an Hilbert space, and $T: H \to H$ is a self-adjoint strictly positive operator (i.e. $\langle Tx,x\rangle > 0$ for all $x \neq 0$). How do I show that this operator is invertible? For example, I want to show that $\langle Tx , x\rangle$ is bounded below by some positive constant (and then I am happy, I know the rest).
Thank you,
Sasha
Best Answer
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 $\langle Tx,x\rangle=1/n$.