Exercise 8, § 4

Let A be a unital Banach algebra, $x\in A$, and B the subalgebra of A generated by $x$. In order that Sp$_A(x)$ be a finite set all of whose elements are poles of the resolvent of $x$, it is necessary and sufficient that B be of finite dimension. (To show that the condition is necessary, if $R(x, \lambda ) =\sum_{i=1}^pF_i(\lambda )a_i$, where $a_i\in A$ and where $F_i$ is a scalar rational function, the expansion of $R(x, \lambda )$ around $\lambda =\infty$ shows that the $x^n$ are linear combinations of the $a_i.$)