Let $T: X \rightarrow X$ be a bounded linear operator on a complex Banach space $X$. Define the spectrum $\sigma(T)$ of $T$. What is an approximate eigenvalue of $T$ ? What does it mean to say that $T$ is compact?

Assume now that $T$ is compact. Show that if $\lambda$ is in the boundary of $\sigma(T)$ and $\lambda \neq 0$, then $\lambda$ is an eigenvalue of $T$. [You may use without proof the result that every $\lambda$ in the boundary of $\sigma(T)$ is an approximate eigenvalue of $T$.]

Let $T: H \rightarrow H$ be a compact Hermitian operator on a complex Hilbert space $H$. Prove the following:

(a) If $\lambda \in \sigma(T)$ and $\lambda \neq 0$, then $\lambda$ is an eigenvalue of $T$.

(b) $\sigma(T)$ is countable.