Remark

For $f=$ Id$_{Sp(u)}$, one has $f(u) =u$ (assertion b) for $k= 1$). The domain of $u$ therefore coincides with the set of $x\in E$ such that the identity function of Sp($u$) belongs to $\mathscr{L}^2$(Sp($u$)$, \mu_x$); in particular it contains the elements $x\in E$ such that the measure $\mu_x$ has compact support.

The following corollary generalizes the corollary to prop. 16 of IV, p. 247 and prop. 17 of IV, p. 248.