Exercise 6, § 4

Let A be a commutative unital Banach algebra.

a) If $j$ is an idempotent of A, one has exp(2$i\pi j$) $= 1$ and exp($i\pi j$) $= 1-2j$. b) Let $p\in A$ be such that exp($p$) $= 1$. Then Sp$_A(p)$ consists of a finite number of points of the form $2in\pi$, where $n\in \mathbf{Z}$.

c) For $\lambda \notin$ Sp$_A(p)$, one has:

$\int_0^1$ exp($t(p-\lambda 1)$)$dt= (1-$ exp($-\lambda$ ))$(\lambda 1-p)^{-1}$. d) Deduce that every point of Sp$_A(p)$ is a simple pole of the resolvent $R(p, \lambda )$, then that there exist an integer $m\geqslant 0$, integers $n_k\in \mathbf{Z}$ and pairwise orthogonal idempotents $j_k$, for $1\leqslant k\leqslant m$, such that