Exercise 28, § 2

Let $A$ be a unital complex Banach algebra, $\Delta$ an open subset of $\mathbf{C}$ and $\lambda \mapsto R(\lambda )$ a mapping from $\Delta$ into $A$ such that

for all $\lambda , \mu\in \Delta$. (a) Let $\lambda_0\in \Delta$. For all $\lambda \in \Delta$ such that $|\lambda -\lambda_0| \cdot \|R(\lambda )\|<1$, one has:

(b) For all $\lambda \in \Delta$ and every integer $n\geqslant 0$, one has

$\frac{\partial^n}{\partial \lambda^n}R(\lambda ) = (-1)^nn$!R($\lambda$ )$^{n+1}$.

(c) If R is holomorphic at infinity, then there exist $z, j, x\in A$ such that $z^2= 0$, $j^2=j,zj=jz= 0,x\in jAj$, and $R(\lambda ) =z+jR(\lambda , x)$ for $|\lambda |$ sufficiently large. (Write $R(\lambda ) =\sum^{\infty}_{n=0}c_n\lambda^{-n}$ for $|\lambda |> \lambda_0$, and express that R satisfies (5). One may put $c_0=z,c_1=j,c_2=x.$)

(d) In order that there exist $a\in A$ such that R is the restriction to $\Delta$ of the resolvent $\lambda \mapsto R(a, \lambda )$ of $a$, it is necessary and sufficient that there exist $\lambda_0\in \Delta$ such that $R(\lambda_0)$ is invertible.

(e) For every $x\in A$, the function $\lambda \mapsto \sum^{\infty}_{n=0}(\lambda_0-\lambda )^nx^{n+1}$ satisfies (5) for $|\lambda -\lambda_0| \cdot \|x\|<1$.