Exercise 5, § 7

Let $s$ be the endomorphism of $\ell^2(\mathbf{N})$ such that

Now prove that the image of $s$ is the open unit disk Δ in $\mathbf{C}$, and that the spectrum of $s$ is the closed unit disk in $\mathbf{C}$. Determine the eigenvalues of $s$. Do analogously for the adjoint $s^*$ of $s$.