SIAM Journal on Matrix Analysis and Applications
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The celebrated Kreiss matrix theorem is one of several results relating the norms of the powers of a matrix to its pseudospectra (i.e., the level curves of the norm of the resolvent). But to what extent do the pseudospectra actually determine the norms of the powers? Specifically, let $A,B$ be square matrices such that, with respect to the usual operator norm $\|\cdot\|$, we have $\|(zI-A)^{-1}\|=\|(zI-B)^{-1}\|$ $(z\in{\mathbb C}).$ (Call this $(*)$.) Then it is known that $1/2\le\|A\|/\|B\|\le 2$. Are there similar bounds for $\|A^n\|/\|B^n\|$ for $n\ge2$? Does the answer change if $A,B$ are diagonalizable? What if $(*)$ holds, not just for the norm $\|\cdot\|$, but also for higher-order singular values? What if we use norms other than the usual operator norm? The answers to all these questions turn out to be negative, and in a rather strong sense.