What are the main properties of eigenvalues of normal unbounded operators?

Question

I am interested in the properties of the eigenvalues of unbounded normal operators. For compact linear operators we have that for every $t >0$, the set of distinct eigenvalues $\lambda$ such that $|\lambda| > t$ is finite. Is there something like this for unbounded normal linear operators? Are there any other easy to understand properties of the eigenvalues of unbounded normal linear operators? For example, are they countable? I hope this question is well-posed. If it is not, please let me know, and I will edit accordingly.

Answer 1

For normal operators on a Hilbert space, eigenvectors for distinct eigenvalues are always orthogonal. Thus if the Hilbert space is separable, there are at most countably many eigenvalues. In a non-separable Hilbert space, this is no longer true. For example, consider $\ell^2(\Gamma)$, the Hilbert space of square-summable functions on an uncountable set $\Gamma$, and the multiplication operator $M_g$ corresponding to an unbounded function $g:\Gamma \to \mathbb C$. Then all members of the range of $g$ are eigenvalues.