Proposition 13

Let $u$ be a positive self-adjoint partial operator on E. Let $q_u$ be the associated positive partial form.

a) The domain of $q_u$ is the domain of $u^{1/2}$, and for all $x$ and $y$ in dom($u^{1/2}$), one has $q_u(x, y) =\langle u^{1/2}(x)|u^{1/2}(y)\rangle$;

b) The positive partial form $q_u$ is closed. The domain of $u$ is a core for $q_u$.

Since $u$ is positive, one has $|u|=u$. The domain dom($|u|^{1/2}$) of $q_u$ therefore coincides with dom($u^{1/2}$), and one has

Sp($u$) Sp($u$)

for $x$ and $y$ in dom($q_u$) (prop. 4, b) of IV, p. 271). This proves assertion a).

Moreover, the prehilbertian space $E_{q_u}$ then coincides with the prehilbertian space $E_{u^{1/2}}$ associated with $u^{1/2}$ (def. 4 of IV, p. 230). Since $u^{1/2}$ is a closed partial operator, this space is a Hilbert space (prop. 4 of IV, p. 230) and dom($u$) is dense in $E_{u^{1/2}}$ by assertion e) of cor. 1 of IV, p. 273.

Let $q$ be a partial hermitian form on E with domain D dense in E. For $y\in D$, let us denote by $\lambda_y$ the linear form $x\mapsto q(y, x)$ on D. We denote by $\widetilde{D}$ the set of $y\in D$ such that the linear form $\lambda_y$ is continuous on D.

Let $y\in \widetilde{D}$. Since D is dense in E, there exists a unique continuous linear form on E which extends $\lambda_y$. We denote it again by $\lambda_y$. There exists a unique element $u(y)$ in E such that $\lambda_y(x) =\langle u(y)|x\rangle$ for every $x\in E$ (EVT, V, p. 15, th. 3). The mapping $y\mapsto u(y)$ is linear from $\widetilde{D}$ into E; the partial operator $u$ on E with domain $\widetilde{D}$ is called the partial operator representing $q$. One therefore has

for $y\in$ dom($u$) and $x\in D$.