Exercise 38, § 5

Let $u$ be a partial operator in a complex Hilbert space E.

a) Suppose that $u$ is a positive symmetric operator. Let $F(u)$ be the Friedrichs extension of $u$ and let $q$ be the positive partial form associated with $F(u)$. Now prove that $F(u)$ is the unique self-adjoint extension of $u$ whose domain is contained in dom($q$).

b) Suppose that $u$ is a closed partial operator with dense domain. Let $q$ be the positive partial form of domain dom($u$) defined by $q(x, y) =\langle u(x)|u(y)\rangle$. Prove that the self-adjoint partial operator representing $q$ is $v=u^*u$.

c) Suppose that $u$ is symmetric and that $u^2$ has dense domain. The partial operator $u^2$ is symmetric and positive; the Friedrichs extension of $u^2$ is equal to $u^*u$.