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 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 ϵ > 0 \epsilon >0 , there are fixed values C > 0 , K > 1 C>0, K>1 and a sequence of ( N + 1 ) × ( N + 1 ) (N+1)\times (N+1) matrices A N A_N , satisfying the resolvent condition, such that ‖ ( A N ) n ‖ ≥ C ( N + 1 ) 1 − ϵ \|(A_N)^n\|\ge C(N+ 1)^{1-\epsilon } = C ( n + 1 ) 1 − ϵ =C(n+1)^{1-\epsilon } for N = n = 1 , 2 , 3 , … N=n=1,2,3,\ldots . The result proved in this paper is also relevant to matrices A A whose ϵ \epsilon -pseudospectra lie at a distance not exceeding K ϵ K\epsilon from the unit disk for all ϵ > 0 \epsilon >0 .