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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.

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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.

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