Exercise 32, § 5

Let E be a complex Hilbert space. Let $u$ and $v$ be partial self-adjoint operators on E such that $u+v$ is self-adjoint.

a) For $t\in \mathbf{R}^*$, let $v(t) =t^{-1}(e^{itu}e^{itv}-e^{it(u+v)})\in \mathscr{L}$ (E), and let $w(t)$ be the restriction of $v(t)$ to the Hilbert space $E_{u+v}$. Let B be the set of continuous linear mappings $w(t)$. The set B is bounded in the Banach space $\mathscr{L}(E_{u+v},E)$. (Use the Banach-Steinhaus theorem.)

b) One has $w(t)\rightarrow 0$ when $t\rightarrow 0$ in $\mathbf{R}^*$, in the space $\mathscr{L}(E_{u+v},E)$ endowed with the topology of compact convergence.

c) Let $x\in E_{u+v}$. One has

lim$_{n\rightarrow+\infty}(e^{itu/n}e^{itv/n})^nx=e^{it(u+v)}x$. (For $n\in \mathbf{N}$, write

and apply the result of the preceding question to the compact image of the mapping $t\mapsto e^{it(u+v)}x$ from $[-1,1]$ into $E_{u+v}.)$