Exercise 9, § 4

Let $A$ be a unital Banach algebra, $x\in A$, U an open neighbourhood of Sp$_A(x)$ having only a finite number of connected components $U_i(i\in I)$, and $f\in \mathscr{O}(U)$. For $f(x) = 0$ to hold, it is necessary and sufficient that the following conditions be satisfied:

(i) for every $i$ such that Sp$_A(x)\cap U_i$ is infinite or contains a point which is not a pole of the resolvent, one has $f|U_i= 0$ ;

(ii) every pole of order $p$ of the resolvent is a zero of order $\geqslant p$ of $f$. (For the necessity of (ii), reason as in Exercise 8.)