Exercise 30, § 2

a) Let $E$ be a complex Banach space and $x:\mathbf{C}-\{1\} \rightarrow E$ an entire function of $1/(\zeta -1)$. We denote by $x(\zeta ) =\sum^{\infty}_{n=0}a_n\zeta^n,x(\zeta ) =\sum^{\infty}_{n=0}b_n\zeta^{-n}$ the expansions of $x$ for $|\zeta |<1,|\zeta |>1$ respectively. We suppose that there exists $\alpha >0$ such that $\|a_n\|=o(n^{\alpha}),\|b_n\|=o(n^{\alpha})$. Then $x(\zeta )$ is a polynomial in $1/(\zeta -1)$ of degree $< \alpha + 1$. (Let $\varepsilon >0$. We have $\|x(\zeta )\|\leqslant \varepsilon (1-r)^{-1-\alpha}$ for $r_{\varepsilon}\leqslant r=|\zeta |<1,\|x(\zeta )\|\leqslant \varepsilon (1-r^{-1})^{-1-\alpha}$ for $1< r\leqslant r'_{\varepsilon}$. Putting $1-\zeta = (\omega +\frac{1}{2})^{-1},\omega = Re^{i\theta}$, now prove that, for $0\leqslant r <1$ and $R\geqslant 1$, one has

$(1-r)^{-1}\leqslant \frac{9}{4}$ cos$R\theta$

and that, for $1< r$ and $R\geqslant 1$,

$(1-r^{-1})^{-1}\leqslant \frac{9}{4}$ cos$R\theta$.

Put $y(\omega ) =x(\zeta )$ and now prove that $R^{-\alpha-1}|$cos$\theta |^{\alpha+1}\|y(Re^{i\theta})\|$ tends to 0 when R tends to $+\infty$, uniformly in $\theta$. Apply then exerc. 29, b).)

Let A be a complex unital Banach algebra, $q\in A$ a quasi-nilpotent element and $x= 1 +q$.

b) In order that $q^N= 0$, it is necessary and sufficient that $\|x^{\pm n}\|=o(n^N)$ when $n$ tends to $+\infty$. (To see that the condition is sufficient, now prove, by applying a), that $R(\lambda , x)$ is a polynomial in $1/(\lambda -1)$ of degree $\leqslant N$. On the other hand, $R(\lambda , x) =$ $\sum^{\infty}_{n=0}q^n(\lambda -1)^{-n-1}$.)

c) Let R be the radical of A and G a subgroup of the group of invertible elements of A. If, for every $x\in G$, one has $\|x^n\|=o(n),\|x^{-n}\|=o(n)$ for $n\rightarrow +\infty$, then the restriction to G of the canonical mapping $A\rightarrow A/R$ is injective.