Exercise 9, § 4

(a) Find Hilbert spaces E and F and partial operators $u$ and $v$ from E into F such that $u$ and $u+v$ have dense domain and $u^*+v^*$ is different from $(u+v)^*$.

(b) Find Hilbert spaces E, F and G and partial operators with dense domain $u$ and $v$ from E into F and from F into G, respectively, such that $v\circ u$ has dense domain and $u^*\circ v^*$ is different from $(v\circ u)^*$.

(c) Find a Hilbert space E and self-adjoint partial operators $u$ and $v$ on E such that $u+v$ has dense domain but is not self-adjoint.