Do the Pseudospectra of a Matrix Determine its Behavior?

Abstract

Let $A$ and $B$ be square matrices. It is shown that the condition $(R) ||(zI-A)^{-1}|| = ||(zI -B)^{-1}||$ for all $z \in \complex$ is equivalent to the condition $(P) ||p(A)|| = ||p(B)||$ for all polynomials $p$ if $|| \cdot ||$ is the Frobenius norm, but not if $|| \cdot ||$ is the 2-norm.