Can someone help me unpack this i.e. provide an explanation or provided a resource that I could read about this(e.g. textbook or online source), my professor wrote this, and I can't seem to find too much about this in my textbook. For example for the first line, it's definitely a few liner explanations of why each eigensubspace is finite dimensional.
If $T$ compact, $\lambda \neq 0$ eigenvalue, then eigen subspace $H_\lambda$ must be finite dimensional.
For any $r > 0$, let $H_r$ be the direct sum of all the eigen subspaces for eigen values $\lambda$ with $|\lambda| \geq r$.
Then $T$ on $H_r$ is invertible(bounded inverse), inverse of norm $\leq \frac{1}{r}$. Implies $H_r$ is finite dimensional.
$T$ carries $H_r$ into itself, and $T^* = T$. Implies $H_r^\perp$ is carried into itself by $T$. $T|_{H_r^\perp}$ is compact. If $||T|_{H_r^\perp}|| \geq r$, then $H^+$ contains an eigenvector for an eigenvalue $\lambda$ with $\lambda \geq r$.
So $||T|_{H_r^\perp}|| < r$, all $H_{r_n}$ are finite dimensional where $r_n \rightarrow 0$. So get a sequence $\{\lambda_n\}$, $|\lambda| \rightarrow 0$. And the eigensubspaces finite dimensional. So $\bigoplus H_{\lambda_n}$ is all of $H$ except for elements in the $\ker$ of $T$, which may be infinite dimensional. Can be $\{0\}$.