Let $H$ be a complex Hilbert space.

(a) Let $T: H \rightarrow H$ be a bounded linear map. Show that the spectrum of $T$ is a subset of $\left\{\lambda \in \mathbb{C}:|\lambda| \leqslant\|T\|_{\mathcal{B}(H)}\right\}$.

(b) Let $T: H \rightarrow H$ be a bounded self-adjoint linear map. For $\lambda, \mu \in \mathbb{C}$, let $E_{\lambda}:=\{x \in H: T x=\lambda x\}$ and $E_{\mu}:=\{x \in H: T x=\mu x\}$. If $\lambda \neq \mu$, show that $E_{\lambda} \perp E_{\mu}$.

(c) Let $T: H \rightarrow H$ be a compact self-adjoint linear map. For $\lambda \neq 0$, show that $E_{\lambda}:=\{x \in H: T x=\lambda x\}$ is finite-dimensional.

(d) Let $H_{1} \subset H$ be a closed, proper, non-trivial subspace. Let $P$ be the orthogonal projection to $H_{1}$.

(i) Prove that $P$ is self-adjoint.

(ii) Determine the spectrum $\sigma(P)$ and the point spectrum $\sigma_{p}(P)$ of $P$.

(iii) Find a necessary and sufficient condition on $H_{1}$ for $P$ to be compact.