Exercise 2, § 6

We retain the notations of the preceding exercise. We suppose that A contains the unit circle $\mathbf{U}$ in $\mathbf{C}$ and that $0\notin A$. We denote by $\varphi_A$ the characteristic function of A.

a) The mapping

is a continuous linear form on B(A).

b) Let $h$ be the finite rank endomorphism of B(A) such that $h(f) =\ell (f)\varphi_A$. The essential spectrum of $u-h$ is distinct from that of $u$. (Now prove that it contains 0.)