Let $H$ be a separable complex Hilbert space.

(a) For an operator $T: H \rightarrow H$, define the spectrum and point spectrum. Define what it means for $T$ to be: (i) a compact operator; (ii) a self-adjoint operator and (iii) a finite rank operator.

(b) Suppose $T: H \rightarrow H$ is compact. Prove that given any $\delta>0$, there exists a finite-dimensional subspace $E \subset H$ such that $\left\|T\left(e_{n}\right)-P_{E} T\left(e_{n}\right)\right\|<\delta$ for each $n$, where $\left\{e_{1}, e_{2}, e_{3}, \ldots\right\}$ is an orthonormal basis for $H$ and $P_{E}$ denotes the orthogonal projection onto $E$. Deduce that a compact operator is the operator norm limit of finite rank operators.

(c) Suppose that $S: H \rightarrow H$ has finite rank and $\lambda \in \mathbb{C} \backslash\{0\}$ is not an eigenvalue of $S$. Prove that $S-\lambda I$ is surjective. [You may wish to consider the action of $S(S-\lambda I)$ on $\left.\operatorname{ker}(S)^{\perp} .\right]$

(d) Suppose $T: H \rightarrow H$ is compact and $\lambda \in \mathbb{C} \backslash\{0\}$ is not an eigenvalue of $T$. Prove that the image of $T-\lambda I$ is dense in $H$.

Prove also that $T-\lambda I$ is bounded below, i.e. prove also that there exists a constant $c>0$ such that $\|(T-\lambda I) x\| \geqslant c\|x\|$ for all $x \in H$. Deduce that $T-\lambda I$ is surjective.