About the sharpness of the stability estimates in the Kreiss matrix theorem
M. N. Spijker, S. Tracogna, Bruno D. Welfert · Mathematics of Computation · 2002
One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices A A under consideration. This so-called resolvent condition is known to imply, for all n ≥ 1 n\ge 1 , the upper bounds ‖ A n ‖ ≤ e K ( N + 1 ) \|A^n\|\le eK(N+1) and ‖ A n ‖ ≤ e K ( n + 1 ) \|A^n\|\le eK(n+1) . Here ‖ ⋅ ‖ \|\cdot \| is the spectral norm, K K is the constant occurring in the resolvent condition, and the order of A A is equal to N + 1 ≥ 1 N+1\ge 1 . It is a long-standing problem whether these upper bounds can be sharpened, for all fixed K > 1 K>1 , to bounds in which the right-hand members grow much slower than linearly with N + 1 N+1 and with n + 1 n+1 , respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each