Proposition 9

Let A be a closed subset of Sp($u$). Let $\varphi_A$ be the characteristic function of A and $p_A=\varphi_A(u)$ the spectral projector of $u$ defined by A. Let us denote by $E_A$ the image of $p_A$. It is a closed subspace of E.

a) The subspace $E_A$ is the space of the $x\in E$ such that the support of the spectral measure of $x$ relative to $u$ is contained in A;

b) If A is bounded in $\mathbf{C}$, then $E_A$ is contained in the domain of $u$;

c) For every $x$ belonging to the domain of $u$, one has $p_A(x)\in$ dom($u$) and $u(p_A(x))\in E_A$, in particular $u(x)\in E_A$ if $x\in$ dom($u$)$\cap E_A$.

For $x$ in E, one denotes by $\mu_x$ the spectral measure of $x$ relative to $u$.

Let us prove a). Let $x$ be an element of $E_A$. Let $z\in \mathbf{C}-$ A and U a relatively compact open neighbourhood of $z$ which does not meet A. For every function $f\in \mathscr{K}(\mathbf{C})$ with support contained in U, one has

Sp($u$) Sp($u$)$\cap A$ according to formula (10). This means that $z$ does not belong to the support of $\mu_x$. Consequently, the support of $\mu_x$ is contained in A.

Conversely, let $x\in E$ be such that the support of $\mu_x$ is contained in A. We have

Sp($u$)$\cap A$ Sp($u$)

(loc. cit.) therefore $\|p_A(x)-x\|^2=\|p_A(x)\|^2- \|x\|^2\leqslant 0$ and consequently $p_A(x) =x$, that is to say $x\in E_A$.

Assertion b) follows from a) and from the remark in IV, p. 273.

Let us prove c). The domain of $u$ is the set of $x\in E$ such that the identity function of Sp($u$) belongs to $\mathscr{L}^2$(Sp($u$)$, \mu_x$) $($loc. cit.). Since $\mu_{p_A(x)}=\varphi_A\cdot \mu_x$ for $x\in E$ (cor. of Prop. 5 of IV, p. 275), one has $p_A(x)\in$ dom($u$) if $x\in$ dom($u$).

Let $x\in$ dom($u$) and $y=p_A(x)$; one has $y\in$ dom($u$)$\cap E_A$ according to c). The spectral measure of $u(y)$ relative to $u$ is $|$Id$_{Sp(u)}|^2\cdot \mu_y($loc. cit.). Since $\mu_y$ has support in A, the same is true of $\mu_{u(y)}$, hence $u(y)\in E_A$ according to a).