Here's the question:
Let $A$ be a positive operator on a (possibly infinite dimensional) Hilbert space. Let $I$ denote the identity operator. Suppose that $A \geq I$, which is to say that $A - I$ is a positive operator. Prove that $A$ is invertible.
I think that this is true, but I haven't been able to find a proof one way or the other. I would like to avoid invoking any heavy machinery (like the spectral theorem) if possible. I would also be interested in a proof that carries over to more general $C^*$ algebras.
Of course, the proof in the case of finite dimensional spaces is fairly obvious, since it suffices to show that the operator has a trivial kernel. Since that does not suffice here, I really have no clue what my next move should be.
Any guidance here would be greatly appreciated.