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This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question. I've read a thesis where there is an example on ergodic mean, where however there is ...

For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$. Does this extend to the (...