Suppose that $T$ is a bounded linear operator on an infinite-dimensional Hilbert space $H$, and that $\langle T(x), x\rangle$ is real and non-negative for each $x \in H$.

(a) Show that $T$ is Hermitian.

(b) Let $w(T)=\sup \{\langle T(x), x\rangle:\|x\|=1\}$. Show that

$$\|T(x)\|^{2} \leqslant w(T)\langle T(x), x\rangle \quad \text { for each } x \in H$$

(c) Show that $\|T\|$ is an approximate eigenvalue for $T$.

Suppose in addition that $T$ is compact and injective.

(d) Show that $\|T\|$ is an eigenvalue for $T$, with finite-dimensional eigenspace.

Explain how this result can be used to diagonalise $T$.