Lemma 9

Let E be a complex Hilbert space and $u$ a normal partial operator on E. Let $\lambda \in$ Sp($u$). Then $\lambda \in$ Sp$_s(u)$ if and only if there exists an open neighbourhood V of $\lambda$ in $\mathbf{C}$ such that the spectral projector $p_V$ of $u$ defined by V is of finite rank.

Let $\lambda \in$ Sp$_s(u)$. There exists an open neighbourhood V of $\lambda$ in $\mathbf{C}$ such that Sp($u$)$\cap V =\{\lambda \}$; then the spectral projector $p_V=p_{\{\lambda\}}$ is of finite rank (Cor. to Prop. 9 of IV, p. 278).

Conversely, suppose that there exists an open neighbourhood V of $\lambda$ such that the spectral projector $p_V$ is of finite rank $n\in \mathbf{N}$. By Cor. to Prop. 10 of IV, p. 279, the intersection $V\cap$ Sp($u$) contains at most $n$ elements, therefore $\lambda$ is isolated in Sp($u$). Hence it is an eigenvalue of $u$ of spectral multiplicity at most $n$ by Cor. to Prop. 9 of IV, p. 278.

In the remainder of this No., we assume that E is infinite-dimensional; the analogues of the results below when E is finite-dimensional are deduced from No.$^o4$ of IV, p. 153.

Let $u$ be a self-adjoint operator bounded below on E. Let $c$ be a lower bound of $u$; the spectrum of $u$ is contained in $[c,+\infty [$ (Prop. 15). Suppose that the essential spectrum of $u$ is not empty. We then denote by $\varrho_e$ the greatest lower bound of the essential spectrum of $u$, so that $\varrho_e\geqslant c$ and $\varrho_e$ is an element of the spectrum of $u$. We put Sp$_h(u) =$ Sp($u$)$\cap [\varrho_e,+\infty [$; this is a closed subset of Sp($u$), hence also of $\mathbf{C}$, such that inf(Sp$_h(u)$) $=\varrho_e$; it is called the upper part of the spectrum of $u$. If the essential spectrum of $u$ is empty, we put Sp$_h(u) =\emptyset$ and $\varrho_e= +\infty$.

The intersection Sp$_b(u) =$ Sp($u$)$\cap [0, \varrho_e[$ is contained in the discrete spectrum of $u$, and its elements are therefore isolated eigenvalues of finite multiplicity; it is said to be the lower part of the spectrum of $u$. Let $E_b$ be the closed subspace of E generated by the eigenspaces corresponding to the eigenvalues $\lambda \in$ Sp$_b(u)$. By the Cor. to Prop. 9 of IV, p. 278 and Prop. 8 of IV, p. 278, the space $E_b$ is the image of the spectral projector $p_{Sp_b(u)}$ defined by Sp$_b(u)$. Since Sp($u$) is the disjoint union of Sp$_b(u)$ and Sp$_h(u)$, the orthogonal complement $E_h$ of $E_b$ is the image of the spectral projector $p_{Sp_h(u)}$ defined by Sp$_h(u)$. If Sp$_b(u)$ is finite, then the space $E_b$ is of finite dimension; the space $E_h$ is then nonzero, since E is assumed to be infinite-dimensional.