Exercise 16, § 4

Let A be a unital Banach algebra, $x\in A$, and let $k$ be an integer $\geqslant 0$. Let $\mathbf{U}$ be the set of complex numbers of modulus 1.

a) Suppose that Sp$_A(x)\subset \mathbf{U}$ and that $\|R(x, \lambda )\|= O((1- |\lambda |)^k)$ when $|\lambda |$ tends to 1. Prove that $\|x^n\|= O(|n|^k)$ when $n$ tends to $\pm \infty$. (For every $\delta >1$, prove that

$(\delta -1)(1-\delta$ )

where M is independent of $n$ and $\delta$. Take $\delta =n/(n-k)$ when $n$ tends to $+\infty ,\delta =n/(n+k)$ when $n$ tends to $-\infty .)$

b) Suppose that $\|x^n\|= O(|n|^k)$ when $n$ tends to $\pm \infty$. Prove that Sp$_A(x)\subset \mathbf{U}$ and that $\|R(\lambda , x)\|= O((1- |\lambda |)^{k+1})$ when $|\lambda |$ tends to 1. (We have $\varrho (x) =\varrho (x^{-1}) = 1$, hence Sp$_A(x)\subset \mathbf{U}$. For $|\lambda |<1$,

Deduce an upper bound for $\|R(\lambda , x)\|$, and reason in an analogous manner if $|\lambda |>1$.)