Exercise 11, § 2

Let E be a Hilbert space.

a) Let $\varrho$ be a unitary representation of the group $\mathbf{R}$ in E. Let D be a dense subspace of E stable under $\varrho$ such that, for every $x\in D$, the mapping $\psi_x:t\mapsto$ $\varrho (t)x$ is differentiable on $\mathbf{R}$. Let $u$ be the partial operator with domain D such that $u(x) =i^{-1}\psi '_x(0)$ for every $x\in D$. Then $u$ is essentially self-adjoint and $\overline{u}$ is the infinitesimal generator of $\varrho$.

b) Let $u$ be a self-adjoint partial operator on E and let D be a dense subspace of the domain of $u$. Suppose that for every $t\in \mathbf{R}$, one has exp($itu$)$(D)\subset D$. Then the space D is dense in $E_u$.

c) Let $u$ be a self-adjoint partial operator on E. Let F be a Hilbert space and $v$ a self-adjoint partial operator on F. The partial operator $u\otimes 1_F+ 1_E\otimes E$ on $E\otimes F$ defines an essentially self-adjoint partial operator on $E\widehat{\otimes}_2F$.