Exercise 19, § 4

Let E be the Hilbert space $\ell_2(\mathbf{N}^*)$. Let $(\alpha_n)_{n\in\mathbf{N}^*}$ be a sequence of real numbers $\geqslant 2$ such that $\alpha_n$ tends to $+\infty$ when $n\rightarrow +\infty$. Put $\beta_n= 1 +\alpha^{-1}_n$ for all $n\in \mathbf{N}^*$.

a) Let $F\subset E$ be the space of sequences $(x_k)_{k\in\mathbf{N}^*}\in E$ such that the series

converges. It is a dense subspace of E.

b) There exists a closed partial operator $u$ on E with domain F such that for every $x= (x_k)_{k\in\mathbf{N}^*}$ in F, one has $u(x) = (y_k)$ where

=

$y_{2k}\beta_k$ 0 $x_{2k}$

for every $k\in \mathbf{N}^*$.

c) Let $u_n$ be the endomorphism

$\beta_n$ 0

of the Hilbert space $\mathbf{C}^2$ endowed with the canonical scalar product. For every $\lambda \in \mathbf{C}$ such that $|\lambda |<1$, the resolvent of $u_n$ at $\lambda$ is defined and one has

lim$_{n\rightarrow+\infty}\|R(u_n, \lambda )\|= 1$.

d) For every $n\in \mathbf{N}^*$ and every $\lambda \in \mathbf{C}$ such that $|\lambda |<\frac{1}{2}$, one has $\|R(u_n, \lambda )\|<1$. e) The open disk with center 0 and radius $\frac{1}{2}$ is contained in the resolvent set of $u$, and one has $\|R(u, \lambda )\|= 1$ for every $\lambda \in \mathbf{C}$ such that $|\lambda |<\frac{1}{2}$.