Theorem 6

Let E be a real or complex Hilbert space and U the unitary group of E.

(i) The set H of Hermitian elements of L(E) is, with the real normed space structure, a closed vector subspace of L(E) admitting a topological supplement.

(ii) The set H' of elements ≥ 0 of GL(E) is a real analytic submanifold of GL(E).

(iii) The restriction to H of the mapping exp is an isomorphism of real analytic manifolds of H onto H'.

(iv) The mapping (h, u) ↦ (exp h)u of H × U into GL(E) is an isomorphism of real analytic manifolds.

Recall that, if x ∈ L(E), x* denotes the adjoint of x. Let H_1 be the set of x ∈ L(E) such that x* = −x. The formula x = $\frac{1}{2}(x + x^*) + \frac{1}{2}(x - x^*)$ proves that, with its real normed space structure, L(E) is the topological direct sum of H and H_1, whence (i).

Suppose that K = C. In the notation of Proposition 17, H' is the set of h ∈ H ∩ A' such that Sp h ⊂ R^*_+. As exp(R) = R^*_+, (ii) and (iii) follow from Proposition 17 and Spectral Theories, Chapter I, § 4, Proposition 8 and § 6, no. 5. The mapping (h, u) ↦ y = (exp h)u of H × U into GL(E) is bijective by Spectral Theories, Chapter I, § 6, Proposition 15. It is real analytic by the above. The mapping y ↦ h = $\frac{1}{2} \log(y y^*)$ is real analytic and so also is the mapping y ↦ u = (exp h)^{-1}y. Hence (iv).

Suppose that K = R. Let Ē be the complexified Hilbert space of E and J the mapping ξ + iη ↦ ξ − iη (for ξ, η in E) of Ē into Ē. Let Ũ, Ũ', Ũ denote the sets defined for Ē as were H, H', U for E. Then H (resp. H', U) is identified with the set of x ∈ Ũ (resp. Ũ', Ũ) such that JxJ^{-1} = x. Properties (ii), (iii) and (iv) then follow easily from (i) and the analogous properties in the complex case.