Is it true that every closed operator on a separable Hilbert H space only has countably many eigenvalues?


Or put the other way around, if I want to ensure that a (not necessarily bounded) linear operator on a separable Hilbert space only has countably many eigenvalues, is closedness (or better said, closability) a sufficient condition?


(By the term eigenvalue, I do not only mean a point in the spectrum of course, but one that actually fulfills $Tx = \lambda x$.)



Countability of eigenvalues of a linear operator
Let $T:\ell^2\rightarrow\ell^2$ be the backwards shift operator, $T(a\_n) = (a\_2,a\_3,\cdots)$. This is a contraction. For any $\lambda\in\mathbb C$, consider the sequence given by $a\_n = \lambda^n$. Thus $(a\_{n+1}) = (\lambda^2,\lambda^3,\cdots) = \lambda(\lambda,\lambda^2,\cdots)$ and so, if $(a\_n)\in\ell^2$, then $(a\_n)$ is an eigenvector of $T$, for eigenvalue $\lambda$. Of course, $\sum\_n |\lambda^n|^2 = \sum\_n |\lambda^2|^n <\infty$ if and only if $|\lambda|<1$.


So even a bounded operator can have a continuum of eigenvectors.


