# Essential selfadjointness preserved under unitarily transfomration?

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.)