Corollary

Tag 02X6, permanent. Cite it as [Bourbaki, Tag 02X6].

Théories spectrales, chapter IV, § 1. Opérateurs compacts sur un espace hilbertien

Let G be a Hilbert space and $v\in \mathscr{L}^c(F; G)$ . Let $n\in I_E\cap I_F$ .

a) Let $r\in \mathbf{R}^*_+$ . We have

\sum_{i=0}^n\alpha_i(v\circ u)^r\leqslant \sum_{i=0}^n\alpha_i(v)^r\alpha_i(u)^r

b) Let $p, q, r\in \mathbf{R}_+^*$ be such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ . Then

(\sum_{i=0}^n\alpha_i(v\circ u)^r)^{1/r}\leqslant (\sum_{i=0}^n\alpha_i(v)^p)^{1/p}(\sum_{i=0}^n\alpha_i(u)^q)^{1/q}

c) Suppose that F = E. For every integer $m\geqslant 2$ , we have

\sum_{i=0}^n\alpha_i(u^m)\leqslant (\sum_{i=0}^n\alpha_i(u)^2)^{m/2}

Let $r\in \mathbf{R}^*_+$ and let $g$ be the function defined on $\mathbf{R}_+$ by $g(x) =x^r$ . The function $g\circ$ exp is convex (Lemma 6), and therefore assertion a) follows from Prop. 11 applied to the function $g$ .

Assertion b) follows from a) and Hölder's inequality (INT, I, Prop. 4).

Let $m\geqslant 2$ be an integer. Apply b) with $r= 1,p=q= 2$ and $v=u^{m-1}$ . We find

\sum_{i=0}^n\alpha_i(u^m)\leqslant (\sum_{i=0}^n\alpha_i(u^{m-1})^2)^{1/2}(\sum_{i=0}^n\alpha_i(u)^2)^{1/2}

Let us prove c) by induction on $m\geqslant 2$ . The preceding inequality establishes the assertion when $m= 2$ . Suppose that $m\geqslant 3$ and that the assertion is valid for $m-1$ ; since one has

\sum_{i=0}^n\alpha_i(u^{m-1})^2\leqslant (\sum_{i=0}^m\alpha_i(u^{m-1}))^2

the above inequality implies

\sum_{i=0}^n\alpha_i(u^m)\leqslant (\sum_{i=0}^n\alpha_i(u^{m-1}))^{1/2}(\sum_{i=0}^n\alpha_i(u)^2)^{1/2}\leqslant (\sum_{i=0}^n\alpha_i(u)^2)^{m/2}

applying the induction hypothesis.

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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