# Are eigenvalues of finite multiplicity isolated?

I am studying the proof of mini-max principle for self adjoint operators on Hilbert spaces and I know that on this situation, isolated elements of the spectrum are eigenvalues. My question is: is true that eigenvalues of finite multiplicity are isolated elements of the spectrum? I could not prove it but the book I am studying "Unbounded Self-adjoint Operators on Hilbert Space" by Konrad Schmüdgen, appears to claim it.

I appreciate any help.

No, it's not true at all. It's quite possible to have eigenvalues embedded in continuous spectrum. For example, consider the operator of multiplication by $$x$$ on $$L^2([0,1],\mu)$$ where $$\mu$$ is the sum of Lebesgue measure on $$[0,1]$$ and a point mass at $$1/2$$. Or somewhat more generally, take a direct sum Hilbert space $$\mathcal H = \mathcal H_1 \oplus \mathcal H_2$$ and self-adjoint operator $$T = T_1 \oplus T_2$$ where $$T_1: \mathcal H_1 \to \mathcal H_1$$ has continuous spectrum on some interval, while $$T_2: \mathcal H_2 \to \mathcal H_2$$ has a point in that interval as an eigenvalue with finite multiplicity.