I am currently doing exercises from Trefthen and Bau's numerical linear algebra. This problem has two parts.
For the first part, I have managed to prove that if a set $S$ of complex numbers is contained in no half-plane disjoint from the origin, there is no polynomial $p(z)=az+1$ for which $ \sup_{z\in S} |p(z)| < 1 $.
I am stuck on the second part. It says let $A$ be a square matrix, not necessarily normal, whose spectrum satisfies the property in the first part (is not contained in any half-plane disjoint from the origin). We are asked to show there is no polynomial $p(z)=az+1$ for which $ \lVert p(A) \rVert < 1 $. Here the norm is the operator norm induced by the Euclidean norm (2-norm of the matrix).
I have no idea how to proceed with the second part as $A$ is given to be general. I do not know much if anything about the norm of a polynomial of a matrix. I would appreciate any and all help with this and thank all helpers.
Hint: For any matrix $M$ with eigenvalue $\lambda$, we have $\|M\| \geq |\lambda|$. What can you say about the eigenvalues of $M = p(A)$?