Exercise 1, § 6

Let $A$ be a compact subset of $\mathbf{C}$ which is equal to the closure of its interior V. We denote $\mathscr{C}(A) =\mathscr{C}(A;\mathbf{C})$. Let B(A) be the vector subspace of $\mathscr{C}(A)$ formed by the functions $f\in \mathscr{C}(A)$ whose restriction to V is holomorphic; it is a Banach subalgebra of $\mathscr{C}(A)$. Let $u\in \mathscr{L}(B(A))$ be the linear mapping such that $u(f)(z) =zf(z)$ for every $z\in A$ and every $f\in B(A)$.

(a) The spectrum of $u$ is equal to A.

(b) For every $z\in V$, the endomorphism $u-z$ of B(A) is a Fredholm endomorphism of index 1.

(c) For every $z\in A$ - V, the endomorphism $u-z$ of B(A) is not a Fredholm endomorphism.

(d) The stable spectrum of $u$ is the boundary of A, the sensitive spectrum of $u$ is empty, and the essential spectrum of $u$ is A.

(e) Suppose that the complement of A in $\mathbf{C}$ is connected. For every $h\in \mathscr{L}^c$(B(A)), the essential spectrum of $u+h$ is equal to A.