Boundedness from below and density of range

Question

Let $H$ be an Hilbert space and $T: H \to H$ linear and bounded operator. Suppose $$ \langle Tx, x \rangle \ge \|x\|^2 \quad \quad \text{ for all } x \in H $$

I can prove this implies injectivity of $T$ and closeness of its range. Is it true that $T$ is surjective? In other words, can I prove its range is also dense?

Answer 1

Your claim will follow from the following Lemma

Let $H$ be an Hilbert space, $T:H\rightarrow H$ be a continuous map. Then, if $T^\star$ is injective with closed range then $T$ is surjective

See proof here A proof that $T^*$ injective with closed range implies $T$ is surjective .

You have $\left<x,T^\star x\right> \geq \|x\|^2$ and so your conclusions on $T$ are also true for $T^\star$ (same proof). Therefore, $T^\star$ is injective and has a closed range. Thus, by the lemma $T$ is surjective.