Exercise 19, § 5

Let $u$ be a self-adjoint partial operator on a Hilbert space E.

a) Let $t\in \mathbf{R}$ and $x\in E$. If the spectral measure of $x$ relative to $u$ has compact support, then one has

$e^{iu}(x) =\sum u^nn($!x),

$n\in \mathbf{N}$

where the series converges in E.

b) The set of vectors $x\in E$ such that the spectral measure of $x$ relative to $u$ has compact support is a dense subspace of E.