Proposition 10

Tag 081F, permanent. Cite it as [Bourbaki, Tag 081F].

Topological Vector Spaces, chapter V, § 4. Some classes of operators in hilbertian spaces

— Suppose K is C. Let $u \in \mathscr{L}(E)$ . Then there exists a unique pair $(h_1, h_2)$ of hermitian endomorphisms of E, such that $u = h_1 + ih_2$ . In order that u is normal, it is necessary and sufficient that $h_1$ and $h_2$ commute.

For, the relation « $u = h_1 + ih_2,\ h_1^* = h_1,\ h_2^* = h_2$ » is equivalent to

\text{« } h_1 = \frac{1}{2}(u + u^*) \text{ and } h_2 = \frac{i}{2}(u^* - u) \text{ »}.

In addition, we have $h_1 h_2 - h_2 h_1 = \frac{i}{2}(uu^* - u^*u)$ . This proves prop. 10.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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