Lemma 3

Let F be a Hilbert space and $u\in \mathscr{L}(E; F)$. Let $n\in \mathbf{N}$.

a) One has $(\widehat{\wedge}^nu)^*=\widehat{\wedge}^n(u^*)$;

b) If F = E and $u$ is hermitian (resp. positive, normal, unitary), then $\widehat{\wedge}^nu$ is hermitian (resp. positive, normal, unitary) ;

c) One has $|\widehat{\wedge}^nu|=\widehat{\wedge}^n|u|$;

d) Let H be a closed subspace of E. The restriction $(\widehat{\wedge}^nu)|\widehat{\wedge}^nH$ of $\widehat{\wedge}^nu$ to $\widehat{\wedge}^nH$ is equal to $\widehat{\wedge}^n(u|H) ($equality in $\mathscr{L}(\widehat{\wedge}^nH;\widehat{\wedge}^nF))$;

e) Suppose F = E. Let H be a closed subspace of E. One has $(\widehat{\wedge}^nu)_{\widehat{\wedge}^nH}=\widehat{\wedge}^n(u_H)$ in $\mathscr{L}(\widehat{\wedge}^nH)$.

Assertion a) results from EVT, V, p. 39, formula (13). It implies at once that $\widehat{\wedge}^nu$ is hermitian (resp. unitary) when $u$ is hermitian (resp. unitary). If $u$ is positive, then $u^{1/2}$ is hermitian, hence so is $\widehat{\wedge}^n(u^{1/2})$; the relation $\widehat{\wedge}^nu=(\widehat{\wedge}^nu^{1/2})^2$ implies that $\widehat{\wedge}^nu$ is positive (I, p. 138, prop. 8). This proves b).

The foregoing makes it possible to calculate that

Since $\widehat{\wedge}^n|u|$ is positive by b), it follows from prop. 16 of I, p. 118 that $\widehat{\wedge}^n|u|=|\widehat{\wedge}^nu|$, whence c).

Let $i_H$ be the canonical injection of H into E. The space $\widehat{\wedge}^nH$ is identified with a closed subspace of $\widehat{\wedge}^nE$ by the mapping $\widehat{\wedge}^ni_H$, whence

and

which proves d) and e).

Let F be a Hilbert space, and let $u\in \mathscr{L}^c(E; F)$. As in IV, p. 151, we denote by $I_E$ the set of dimensions of finite-dimensional subspaces F of E such that $F\not = E$. Recall that $(\alpha_n(u))_{n\in I_E}$ denotes the extended sequence of singular values of $u$ (def. 3 of IV, p. 156).