Rudin - Functional analysis excercise 12.7

Question

Suppose $U \in \mathcal{B}(H)$ is unitary, and $\epsilon > 0$. Prove that scalars $\alpha_{0}, ..., \alpha_{n}$ can be chosen so that \begin{align*} \| U^{-1} - \alpha_{0}I - \alpha_{1}U - ... - \alpha_{n}U^{n} \| < \epsilon \end{align*} if $\sigma(U)$ is a proper subset of the unit circle, but that this norm is never less than $1$ if $\sigma(U)$ covers the whole circle. For the first case, I have tried considering expressing $U$ in terms of $e^{iV}$ where $V$ is self adjoint, but can not proceed any further..Please Help!!!!!

Answer 1

This is immediate from the spectral theorem, plus two exercises in elementary complex analysis: (i) If $K$ is a proper compact subset of the unit circle then there exist polynomials $P_n$ such that $P_n(z)\to1/z$ uniformly on $K$, (ii) If $P$ is a polynomial then the sup of $|P(z)-1/z|$ over the unit circle is at least $1$.