Proposition 16

The set Sp$_b(u)\subset [c, \varrho_e[$ is countable, discrete and closed in $[0, \varrho_e[$. It is the set of values of a strictly increasing sequence $(\nu_k)_{0\leqslant k<Card(Sp_b(u))}$ of positive real numbers; if Sp$_b(u)$ is infinite, then the sequence $(\nu_k)$ converges to $\varrho_e$.

Let us set T = Sp$_b(u)\cap [0, \varrho_e[$. It is a closed and discrete subset of $[c, \varrho_e[$ by definition. For every integer $i\geqslant 1$, the set Sp$_b(u)\cap [c, \varrho_e-1/i]$ is compact and discrete, hence finite. Since T is the union of these sets for $i\geqslant 1$, the set T is countable.

This concludes the proof if Sp$_b(u)$ is finite. If Sp$_b(u)$ is infinite, one concludes by applying Lemma 2 of III, p. 90 to the image S of T by the mapping $\lambda \mapsto \varrho_e-\lambda$.

For every integer $k$ such that $0\leqslant k <$ Card(Sp$_b(u)$), let $n_k\geqslant 1$ denote the multiplicity of the eigenvalue $\nu_k$ of $u$. Let us set $\mathbf{N}=\mathbf{N}\cup \{+\infty \} \subset \mathbf{R}$. Let $M\in \overline{\mathbf{N}}$ be the sum of the multiplicities $n_k$. It is the Hilbert dimension of $E_b$, if one agrees to say that a space of infinite Hilbert dimension is of dimension $+\infty \in \overline{\mathbf{N}}$.

For every integer $n$ such that $0\leqslant n <M$, one defines $\lambda_n(u) =\nu_k$, where $k\geqslant 0$ is the unique integer such that

If $n\in \mathbf{N}$ satisfies $n\geqslant M$, one sets $\lambda_n(u) =\varrho_e$. This case can occur only if Sp$_b(u)$ is finite, in which case $\varrho_e$ is finite, since it is assumed that E is of infinite dimension.

By construction, the sequence $(\lambda_n(u))_{n\in\mathbf{N}}$ is increasing and, for every element $\lambda$ of Sp$_b(u)$, the number of integers $n$ such that $\lambda_n(u) =\lambda$ is equal to the multiplicity of $\lambda$ as an eigenvalue of $u$. The sequence $(\lambda_n(u))_{n\in\mathbf{N}}$ tends to $+\infty$ if and only if the essential spectrum of $u$ is empty.

For every $n\in \mathbf{N}$, let $\mathscr{F}_n$ (resp. $\mathscr{F}_n^u$) denote the set of vector subspaces $F\subset E$ of dimension $n$ (resp. the set of vector subspaces $F\subset$ dom($u$) of dimension $n$).

Let $n\in \mathbf{N}$ be such that $n <M$ and let $F\in \mathscr{F}_n^u$. One says that F is adapted to $u$ if F admits an orthonormal basis $(f_i)_{0\leqslant i\leqslant n-1}$ such that $u(f_i) =\lambda_i(u)f_i$ for $0\leqslant i\leqslant n-1$.

Let $F\in \mathscr{F}_n^u$ be adapted to $u$. The space F is contained in $E_b$; it is generated by eigenvectors of $u$ for eigenvalues $\lambda \leqslant \lambda_{n-1}(u)$, and it contains the eigenspaces corresponding to the eigenvalues $\lambda < \lambda_{n-1}(u)$. Moreover, for every universally measurable subset A of Sp($u$), the space F is stable under the spectral projector $p_A$ defined by A (prop. 9, c) of IV, p. 278).