Exercise 19, § 2

Let $X\subset \mathbf{R}$ be a compact subset of $\mathbf{R}$ and $\mu$ a positive measure on X, and let $u$ be the endomorphism of multiplication by the identity function of X.

Put $E = L^2(X, \mu)$ and denote by $\mathscr{L}_s(E)$ the space of endomorphisms of E endowed with the topology of simple convergence.

Let $\varepsilon >0$. Let $r >0$ be such that $X\subset ]-r, r[$, and let $\ell \geqslant 1$ be such that $r2^{-\ell}< \varepsilon$. For every integer $n\in \mathbf{N}$ and every integer $j$ such that $|j|\leqslant 2^{n+\ell}$, let

a) For every integer $n\in \mathbf{N}$, the family $\mathscr{X}_n= (X_{n,j})_j$ is a finite partition of X; the sets $X_{n,j}$ are measurable and of diameter $\leqslant r2^{-n-\ell}$. For every $n\in \mathbf{N}$, the partition $\mathscr{X}_{n+1}$ is finer than the partition $\mathscr{X}_n$. We can therefore apply to $(\mathscr{X}_n)$ the results of the preceding exercise, taking up again its notation, in particular the spaces $E_n$ and $F_n$ and the orthoprojectors $p_n$ and $q_n$. The spaces $E_n$ are finite-dimensional.

b) The series

converges in the space $\mathscr{L}_s(E)$; there exists an orthonormal basis B of E such that $v\in \mathscr{D}_B(E)$.

c) The endomorphism $w=v-u$ is compact. (Consider the endomorphisms $u_n$ of multiplication by the functions $g_n\in E_n$ such that, for every integer $j$ such that $|j|\leqslant 2^{n+\ell}$, and for every $x\in X_{n,j}$, one has $g_n(x) =rj2^{-n-\ell}$; then verify that $\|u-u_n\|\leqslant r2^{-n-\ell}$, and that $u_np_n=p_nu_n.$)