Exercise 5, § 4

Let $A$ be a unital Banach algebra and $a\in A$. Let G be the group of invertible elements of A and $G_1$ the neutral component of G. The exponential spectrum of A is the set of $\lambda \in \mathbf{C}$ such that $x-\lambda e$ does not belong to $G_1($cf. prop. 12 of I, p. 79).

a) The exponential spectrum of $a$ is closed; it contains the spectrum of $a$.

b) The boundary of the exponential spectrum of $a$ is contained in the singular spectrum of $a$ (exercise 16 of I, p. 158).

c) The exponential spectrum of $a$ is the union of the spectrum of $a$ and of certain bounded connected components of $\mathbf{C}-$ Sp($a$).

d) Let $f\in \mathbf{C}[X]$ be a polynomial. The image by $f$ of the exponential spectrum of $a$ contains the exponential spectrum of $f(a)$. In general, the inclusion is strict (use Fredholm theory, cf. III, to appear).