Exercise 12, § 5

One equips $\mathbf{R}$ with Lebesgue measure. Let E be the closed subspace of $L^2(\mathbf{R})$ formed by the $f\in L^2(\mathbf{R})$ such that $f(-x) =f(x)$ for almost every $x\in \mathbf{R}$. Let $\mathscr{D}_+(\mathbf{R})$ be the space of test functions on $\mathbf{R}$ such that $f(x) =f(-x)$ for all $x\in \mathbf{R}$. We identify $\mathscr{D}_+(\mathbf{R})$ with a subspace of E. We denote by $\mathscr{F}\in \mathscr{L}(E)$ the endomorphism of E deduced from the Fourier transform of $L^2(\mathbf{R})$ by passing to the subspaces.

a) For every $f\in \mathscr{D}_+(\mathbf{R})$, one has $\mathscr{F}(f) +\mathscr{F}^*(f)\in \mathscr{D}_+(\mathbf{R})$. b) Let $u$ be the partial operator on E whose domain is $\mathscr{D}_+(\mathbf{R})$ and which is defined by $u(f) =\mathscr{F}(f) +\mathscr{F}^*(f)$ for $f\in \mathscr{D}_+(\mathbf{R})$. The operator $u$ is essentially self-adjoint.

c) One has dom($u^2$) $=\{0\}$. (Use Exercise 28 of II, p. 273.)