Exercise 6, § 5

Let E and F be Hilbert spaces. Let $u$ be a closed partial operator with dense domain from E into F. Let $\lambda \in \mathbf{C}^*$ belonging to the resolvent set of the self-adjoint operator $u^*\circ u$ on E (cf. Prop. 12 of IV, p. 241).

a) Show that the partial operator $u\circ u^*$ from F into F is self-adjoint.

b) Show that for every vector $x\in$ dom($u^*$), the vector

belong to dom($u\circ u^*$), and that the formula

holds.

c) Show that the partial operator $u\circ R(u^*\circ u, \lambda )\circ u^*$ has dense domain, and is continuous on its domain.

d) Show that the partial operator $u\circ u^*-\lambda 1_F$ is surjective.

e) Show that $\lambda \notin$ Sp($u\circ u^*$).

f) Prove the identity

in $\mathscr{L}(F)$ (where $u\circ R(u^*\circ u, \lambda )\circ u^*$ denotes, by abuse of notation, the endomorphism of F which coincides with this partial operator on its domain, cf. question c)).

g) Show that one has the relation

h) Show that the partial operator $u$ defines by passing to subspaces an isomorphism of Ker($u^*\circ u-\lambda 1_E$) onto Ker($u\circ u^*-\lambda 1_F$).