Proposition 18

Let $u$ be a self-adjoint partial operator on E and $\lambda$ a real number. Then $\lambda \in$ Sp$_e(u)$ if and only if there exists an orthonormal sequence $(x_n)_{n\in\mathbf{N}}$ in E such that $u(x_n)-\lambda x_n$ tends to 0 in E.

Suppose first that $\lambda \in$ Sp$_e(u)$. If the spectral projector of $u$ relative to $\{\lambda \}$ is of infinite rank, any orthonormal sequence $(x_n)$ in its image answers the question (cf. cor. to prop. 9 of IV, p. 278). We shall suppose in the sequel that this is not the case.

For each $k\in \mathbf{N}$, let $J_k$ be the set of $t\in [\lambda -1, \lambda + 1]$ such that

The sets $J_k$ are pairwise disjoint. Moreover, for every integer $K\in \mathbf{N}$, the sets $(J_k)_{k\geqslant K}$ form a partition of the set

Since the spectral projector of $u$ relative to $I_K\cup \{0\}$ is of infinite rank (lemma 9 of IV, p. 297) and that relative to $\{\lambda \}$ is supposed to be of finite rank, one deduces from prop. 8 of IV, p. 278 that there exists a strictly increasing sequence $(k_n)_{n\in\mathbf{N}}$ in $\mathbf{N}$ such that the spectral projector $p_n$ of $u$ relative to $J_{k_n}$ is nonzero. Let $x_n$ be a vector of norm 1 in the image of $p_n$. The sequence $(x_n)$ is orthonormal, since the image of $p_n$ is orthogonal to that of $p_m$ for all $n\not =m$ in $\mathbf{N}$.

Let $n\in \mathbf{N}$. Let $\mu_n$ denote the spectral measure of $x_n$ relative to $u$; its support is contained in $J_{k_n}$ (prop. 9 of IV, p. 278). It follows that

therefore the sequence $(x_n)_{n\in\mathbf{N}}$ has the required property.

Conversely, suppose that there exists an orthonormal sequence $(x_n)_{n\in\mathbf{N}}$ in E such that $u(x_n)-\lambda x_n\rightarrow$ 0. Let $\mu_n$ denote the spectral measure of $x_n$ relative to $u$.

Let $\varepsilon >0$. Let $p_{\varepsilon}$ denote the spectral projector of $u$ relative to the open interval $I_{\varepsilon}= ]\lambda -\varepsilon , \lambda +\varepsilon [$. For every $n\in \mathbf{N}$, we have

The assumption on the sequence $(x_n)$ therefore implies that the norm of $p_{\varepsilon}(x_n)$ cannot tend to 0. Since the orthonormal sequence $(x_n)$ converges weakly to 0 in E (Lemma 13), the projector $p_{\varepsilon}$ cannot be compact (Cor. of Prop. 6 of III, p. 6) and is consequently of infinite rank. Since this holds for every $\varepsilon >0$, Lemma 9 of IV, p. 297 allows us to conclude that $\lambda \in$ Sp$_e(u)$.

The following theorem is the analogue for self-adjoint partial operators of Theorem 3 of III, p. 93.