Exercise 1, § 4

Soit $n\geqslant 1$ an integer and $A =\mathscr{L}(\mathbf{C}^n)$. Let $\alpha \in \mathbf{C}$ and let $x\in A$ be an element whose matrix relative to the canonical basis of $\mathbf{C}^n$ is a Jordan matrix of order $n$ and of eigenvalue $\alpha$ (A, VII, p. 34, def. 3). The spectrum of $x$ is reduced to $\alpha$.

a) For $f\in \mathscr{O}(\{\alpha \})$, calculate the matrix relative to the canonical basis of $\mathbf{C}^n$ which represents $f(x)$.

Let $m\geqslant 1$ be an integer. Let $B_m$ be the unital Banach algebra of jets of functions of class $C^m$ in a neighbourhood of $\alpha$ endowed with the topology of uniform $C^m$-convergence (VAR, R2, §12).

b) There exists a continuous morphism of unital algebras from $B_{m-1}$ into A such that $\varphi (z) =x$, where $z\in B_{m-1}$ is the jet of order $m-1$ at $\alpha$ of the identity function of $\mathbf{C}$.

c) There does not exist a continuous morphism $\varphi$ of unital algebras from $B_m$ into A such that $\varphi (z) =x$, where $z\in B_m$ is the jet of the identity function of $\mathbf{C}$.