About the sharpness of the stability estimates in the Kreiss matrix theorem

Abstract:

One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices $A$ under consideration. This so-called resolvent condition is known to imply, for all $n\ge 1$, the upper bounds $\|A^n\|\le eK(N+1)$ and $\|A^n\|\le eK(n+1)$. Here $\|\cdot \|$ is the spectral norm, $K$ is the constant occurring in the resolvent condition, and the order of $A$ is equal to $N+1\ge 1$. It is a long-standing problem whether these upper bounds can be sharpened, for all fixed $K>1$, to bounds in which the right-hand members grow much slower than linearly with $N+1$ and with $n+1$, respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each $\epsilon >0$, there are fixed values $C>0, K>1$ and a sequence of $(N+1)\times (N+1)$ matrices $A_N$, satisfying the resolvent condition, such that $\|(A_N)^n\|\ge C(N+ 1)^{1-\epsilon }$ $=C(n+1)^{1-\epsilon }$ for $N=n=1,2,3,\ldots$. The result proved in this paper is also relevant to matrices $A$ whose $\epsilon$-pseudospectra lie at a distance not exceeding $K\epsilon$ from the unit disk for all $\epsilon >0$.