Exercise 35, § 5

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

a) For every $t\in \mathbf{R}^*_+$, one has $u\circ R(u,-t)\in \mathscr{L}(E)$.

b) The mapping $f$ of $\mathbf{R}_+^*$ into $\mathscr{L}(E)$ defined by $f(t) =-t^{-1/2}u\circ R(u,-t)$ is integrable with respect to the Lebesgue measure on $\mathbf{R}^*_+$.

c) One has