Exercise 1, § 7

Let $E =\mathscr{C}([0,1])$ and $u$ the endomorphism of E defined by

a) The spectral radius of $u$ is zero. (Estimate the norm of $u^n.$)

b) The spectrum of $u$ is reduced to 0, and for every $n\geqslant 1$, the kernel of $u^n$ is reduced to 0.

(In particular, the point 0 is an isolated point of the spectrum of $u$, but the closure of the union of the kernels of $u^n$ for $n\geqslant 1$ is not equal to the spectral subspace $E_0(x)$.)