Exercise 29, § 2

a) Let E be a complex Banach space, $f:\mathbf{C}\rightarrow E$ an entire function and $\theta \mapsto P(\theta ) =\sum^m_{\mu=k}c_\mu e^{\mu i\theta}$ a trigonometric polynomial $(c_k\not= 0$). It is assumed that $|P(\theta )| \cdot \|f(re^{i\theta})\|\leqslant Mr^{\alpha}$ for $r\geqslant r_0($M$,r_0,\alpha$ being constants $\geqslant 0$). Then $f$ is a polynomial of degree $\leqslant \alpha$. (Let $f(\zeta ) =$ $\sum^{\infty}_{\nu=0}a_{\nu}\zeta^{\nu}$, and

Calculate the limit of $J(r, p)$ when $r\rightarrow +\infty$ in two different ways for $p > \alpha .)$

b) With the notation of a), if $\alpha$ is an integer and if, for every $\theta \in \mathbf{R}$, one has

lim$_{r\rightarrow+\infty}r^{-\alpha}|P(\theta )| \|f(re^{i\theta}) = 0$,

then $f$ is a polynomial of degree $< \alpha$.

c) Let A be a complex unital Banach algebra and $x$ an element of A which is quasi-nilpotent. It is assumed that, for a certain integer $n >0$, and for every $\theta \in \mathbf{R}$, one has

lim$_{r\rightarrow+\infty}r^{-n}|$ cos$^n\theta | \cdot \|(1-re^{i\theta}x)^{-1}\|= 0$.

Then $x^n= 0$. (Use b).)