Definition 8

Let D be a vector subspace of E. A partial Hermitian form on E with domain D is a correspondence $q= (\Gamma ,E\times E,\mathbf{C})$ whose domain of definition is $D\times D$, whose graph Γ is functional, and such that the mapping from $D\times D$ into $\mathbf{C}$ defined by Γ is a Hermitian form. We write dom($q$) $= D$.

We say that $q$ is a positive partial form if the Hermitian form which it defines is positive.

Let $u$ be a self-adjoint partial operator on E. For every elements $x$ and $y$ of E, we denote by $\mu_{x,y}$ (resp. $\mu_x$) the spectral measure of $(x, y)$ (resp. of $x$) relative to $u$.

Let $(j,|u|)$ be the polar decomposition of $u$ (Def. 7 of IV, p. 290). Let us denote by $D'$ the domain of $|u|^{1/2}$. By definition, this is the space of the $x\in E$ such that the function $z\mapsto |z|$ is integrable on Sp($u$) with respect to the measure $\mu_x$. It contains dom($u$).

Let $(x, y)\in D'\times D'$. The identity function on Sp($u$) is integrable with respect to the measure $\mu_{x,y}$ (Prop. 4, b) of IV, p. 271); we put

Sp($u$)

If $y\in$ dom($u$), we have $q_u(x, y) =\langle x|u(y)\rangle$ by formula (6) of IV, p. 271. The mapping $q_u$ is a Hermitian form on $D'$. It defines a partial Hermitian form on E, said to be associated with $u$. If the operator $u$ is positive, then the form $q_u$ is a positive partial Hermitian form.