Remark 4

Suppose $K =\mathbf{C}$. Let $u\in \mathscr{P}(E; E)$ be a partial operator with dense domain. The condition $\langle x|u(x)\rangle \in \mathbf{R}$ for every $x\in$ dom($u$) implies that $u$ is symmetric (EVT, V, p. 2, remark); in particular, if $\langle x|u(x)\rangle \in \mathbf{R}_+$ for every $x\in$ dom($u$), then $u$ is positive.