Remark 7

Tag 033E, permanent. Cite it as [Bourbaki, Tag 033E].

Théories spectrales, chapter IV, § 4. Opérateurs partiels

Let $u$ and $v$ be symmetric partial operators on E. If $u+v$ has dense domain, for example if $u$ or $v$ belong to $\mathscr{L}$ (E), then $u+v$ is symmetric. In general, the partial operator $u+v$ is not self-adjoint, even if $u$ and $v$ are (exercise 9 of IV, p. 347).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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