Remark 1

For an operator $u$ with dense domain on E to be symmetric, it is necessary and sufficient that one have

for every $(x, y)\in$ dom($u$)$^2$. This formula in fact shows that the domain of $u$ is contained in that of $u^*$, and then that $u^*$ and $u$ coincide on the domain of $u$. In particular, it follows that $\langle x|u(x)\rangle \in \mathbf{R}$ for every $x\in$ dom($u$).

As will be seen in different examples, formula (2) can often be verified by a straightforward calculus. On the other hand, the exact determination of the domain of the adjoint, which alone makes it possible to know whether or not a symmetric operator is self-adjoint, may be very delicate.