Exercise 18, § 4

Let E be the complex vector space of bounded sequences $(x_k)_{k\in\mathbf{Z}}$ of complex numbers. For every $x= (x_k)$ in E, we write

$\|x\|_E=|x_0|+$ sup$_{k\in\mathbf{Z}-\{0\}}|x_k|$.

a) The mapping $x\mapsto \|x\|_E$ is a norm on E; the space E endowed with this norm is a Banach space.

b) Let $M>2$ be a real number. Let $\lambda = (\lambda_k)_{k\in\mathbf{Z}}$ be the sequence defined by $\lambda_0= M^{-1}$ and $\lambda_k$ = 1 if $k\not = 0$. We denote by $u$ the linear mapping of E into E such that $u((x_k)_{k\in\mathbf{Z}}) = (\lambda_kx_{k+1})_{k\in\mathbf{Z}}$. It is a continuous endomorphism of E.

c) The open disk with center 0 and radius $1/M$ is contained in the resolvent set of $u$.

d) For every $\lambda \in \mathbf{C}$ such that $|\lambda |<$ inf(M$^{-1},\frac{1}{2}-M^{-1}$), one has $\|R(u, \lambda )\|\leqslant M$. (Use formula (6) of IV, p. 244).) e) Let $e_0\in E$ be the sequence $(x_k)_{k\in\mathbf{N}}$ defined by $x_0= 1$ and $x_k= 0$ if $k\not = 0$. Let $\lambda \in \mathbf{C}$ such that $|\lambda |<$ inf(M$^{-1},\frac{1}{2}-M^{-1}$). One has $\|R(u, \lambda )e_0\|_E= M$, hence $\|R(u, \lambda )\|= M$.

(This example, like that of the following exercise, is due to E. Shargorodsky, « On the level sets of the resolvent norm of a linear operator », Bull. London Math. Soc. 40 (2008), 493–504.)