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I am wondering if essential selfadjointness of an operator in a Hilbert space is preserved under unitarily transformations.

In other terms: let $H,H'$ be two isomorphic Hilbert spaces, with an isomorphism between the two denoted by $U$. Let $A$ be an operator in $H$ with domain $D$ and consider the operator $A':= U A U^{-1}$ in $H'$ with domain $D':= U(D)$.

Is it true that $A$ is essentially selfadjoint if and only if $A'$ is?

(Essential selfadjoitness means that the operator is symmetric and its closure is selfadjoint. Equivalently the operator is symmetric and has a unique selfadjoint extension.)

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The properties you are interested in are defined in terms of inner products and norm limits. These are preserved by unitaries, so your property has to be preserved.

In (even more) "lay terms", a unitary is nothing else than an isomorphism of Hilbert spaces; and an isomorphism is nothing else than a "re-labelling" of the elements.

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