Exercise 16, § 2

Let $A$ be a unital complex Banach algebra. Let $a$ be an element of A. The singular spectrum of A is called the set of $\lambda \in \mathbf{C}$ such that $a-\lambda e$ is a topological zero divisor on the left and right in A.

(a) Prove that the singular spectrum of A is contained in the spectrum of A, and contains the boundary of the spectrum of A.

(b) Let $f\in \mathbf{C}[X]$ be a polynomial. The image by $f$ of the singular spectrum of $a$ is the singular spectrum of $f(a)$.