Definition 7

Let $u$ be a partial operator on E. One says that $u$ is symmetric if $u$ has dense domain and if $u^*$ is an extension of $u$. One says that $u$ is self-adjoint if $u$ has dense domain and $u^*=u$. One says that $u$ is essentially self-adjoint if it is closable and if the closure $\overline{u}$ of $u$ is self-adjoint.

One says that $u$ is a partial operator bounded below if $u$ is symmetric and if there exists a real number $c$ such that $\langle x|u(x)\rangle \geqslant c\|x\|^2$ for every $x$ belonging to the domain of $u$. One then says that $c$ is a lower bound of $u$. If $c= 0$, one also says that $u$ is a positive partial operator.

We denote by $\mathscr{A}(E)$ the set of self-adjoint partial operators on E.