Let $H$ be a complex Hilbert space with orthonormal basis $\left(e_{n}\right)_{n=-\infty}^{\infty}$ Let $T: H \rightarrow H$ be a bounded linear operator. What is meant by the spectrum $\sigma(T)$ of $T$ ?

Define $T$ by setting $T\left(e_{n}\right)=e_{n-1}+e_{n+1}$ for $n \in \mathbb{Z}$. Show that $T$ has a unique extension to a bounded, self-adjoint linear operator on $H$. Determine the norm $\|T\|$. Exhibit, with proof, an element of $\sigma(T)$.

Show that $T$ has no eigenvectors. Is $T$ compact?

[General results from spectral theory may be used without proof. You may also use the fact that if a sequence $\left(x_{n}\right)$ satisfies a linear recurrence $\lambda x_{n}=x_{n-1}+x_{n+1}$ with $\lambda \in \mathbb{R}$, $|\lambda| \leqslant 2, \lambda \neq 0$, then it has the form $x_{n}=A \alpha^{n} \sin \left(\theta_{1} n+\theta_{2}\right)$ or $x_{n}=(A+n B) \alpha^{n}$, where $A, B, \alpha \in \mathbb{R}$ and $0 \leqslant \theta_{1}<\pi,\left|\theta_{2}\right| \leqslant \pi / 2$.]