Exercise 8, § 5

Let $u$ and $v$ be normal partial operators on a Hilbert space E. The following assertions are equivalent:

a) The endomorphisms $b(u)$ and $b(v)$ commute;

b) For every universally measurable subset A of $\mathbf{C}$, the spectral projections of $u$ and $v$ on A commute;

If the resolvent set of $u$ and $v$ is nonempty, these assertions are equivalent to

c) For every $\lambda \notin$ Sp($u$) and $\lambda '\notin$ Sp($v$), the resolvents $R(u, \lambda )$ and $R(v, \lambda ')$ commute.

If $u$ and $v$ are self-adjoint, these assertions are equivalent to

d) For every $t\in \mathbf{R}$, the unitary endomorphisms $e^{itu}$ and $e^{itv}$ commute.