Proposition 8

Let $u$ be a positive compact endomorphism of E. Let H be a closed subspace of E and $i_H: H\rightarrow E$ the canonical injection. Denote by $u_H$ the endomorphism $i^*_Hui_H$ of H. It is compact and positive.

a) One has $I_H\subset I_E$ and $\lambda_n(u_H)\leqslant \lambda_n(u)$ for all $n\in I_H$;

b) If H is of finite codimension $k\in \mathbf{N}$ in E, then one has $I_H+k\subset I_E$ and $\lambda_{n+k}(u)\leqslant \lambda_n(u_H)$ for all $n\in I_H$.

The endomorphism $u_H$ is compact (prop. 3 of III, p. 5). It is positive since $\langle x|u_H(x)\rangle =\langle i_H(x)|u(i_H(x))\rangle \geqslant 0$ for all $x\in H$.

Let $n\in I_H\subset I_E$. Let F be a subspace of dimension $n+ 1$ of H adapted to $u_H$. One has therefore $\lambda_n(u_H) =r_F(u_H)$ (prop. 5 of IV, p. 153, a)), and since moreover $r_F(u_H) =r_F(u)\leqslant \lambda_n(u)$ (prop. 6 of IV, p. 154), one obtains assertion a).

Suppose that H is of codimension $k\in \mathbf{N}$ in E and that $n\in I_H$. Let F be a subspace of H of dimension $n$ adapted to $u_H$. Its orthogonal complement in H is equal to $H\cap F^{\circ}$, and it is the orthogonal complement in E of the subspace $F + H^{\circ}$ of dimension $n+k$. Hence $n+k\in I_E$ and (prop. 5 of IV, p. 153, b))

$\lambda_n(u_H) =$ sup $\frac{\langle x|u_H(x)\rangle}{2}= R_{F+H^{\circ}}(u)$

$x\not =0$

whence $\lambda_n(u_H)\leqslant \lambda_{n+k}(u)$ (prop. 6 of IV, p. 154).