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The Wikipedia article on spectral decomposition, see here https://en.wikipedia.org/wiki/Self-adjoint_operator says the following: A self-adjoint operator A on $H$ has pure point spectrum if and ...

Let $D$ be an unbounded densely defined operator on a separable Hilbert space $H$. If $D$ is diagonalisable with all eigenvalues having finite multiplicity and growing towards infinity, does it follow ...

Let $H$ be a Hilbert space and $D:\mathrm{Dom}(D) \to H$ a densely defined operator on $H$. We further assume that $D$ is closed and self-adjoint. If we further assume that $D$ is positive, then we ...