Let $H$ be a Hilbert space and let $T \in \mathcal{B}(H)$.

(a) Define what it means for $T$ to be (i) invertible, and (ii) bounded below. Prove that $T$ is invertible if and only if both $T$ and $T^{*}$ are bounded below.

(b) Define what it means for $T$ to be normal. Prove that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for all $x \in H$. Deduce that, if $T$ is normal, then every point of Sp $T$ is an approximate eigenvalue of $T$.

(c) Let $S \in \mathcal{B}(H)$ be a self-adjoint operator, and let $\left(x_{n}\right)$ be a sequence in $H$ such that $\left\|x_{n}\right\|=1$ for all $n$ and $\left\|S x_{n}\right\| \rightarrow\|S\|$ as $n \rightarrow \infty$. Show, by direct calculation, that

$$\left\|\left(S^{2}-\|S\|^{2}\right) x_{n}\right\|^{2} \rightarrow 0 \quad \text { as } n \rightarrow \infty$$

and deduce that at least one of $\pm\|S\|$ is an approximate eigenvalue of $S$.

(d) Deduce that, with $S$ as in (c),

$$r(S)=\|S\|=\sup \{|\langle S x, x\rangle|: x \in H,\|x\|=1\}$$