The Hilbert-Schmidt Theorem

Let $T \in \mathcal{L}(H)$ be compact and symmetric. Define

Show that if $\alpha < 0$, then $\alpha$ is an eigenvalue of $T$ and if $\beta > 0$, then $\beta$ is an eigenvalue of $T$. Exhibit an example where $\alpha = 0$ and yet $\alpha$ is not an eigenvalue of $T$, that is, $T$ is one-to-one.

Let $K \in \mathcal{L}(H)$ be compact and symmetric. Suppose

Let $\left\{ h_{n}\right\}$ be a sequence of unit vectors for which $\lim_{n \rightarrow \infty } \left\langle K\left(h_{n}\right), h_{n} \right\rangle = \beta$. Show that a subsequence of $\left\{ h_{n}\right\}$ converges strongly to an eigenvector of $T$ with corresponding eigenvalue $\beta$.