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

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 ≥ 1, the upper bounds ||Aⁿ|| ≤ eK(N + 1) and ||Aⁿ|| ≤ eK(n + 1). Here K is the spectral norm, K is the constant occurring in the resolvent condition, and the order of A is equal to N + 1 ≥ 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 ∈ > 0, there are fixed values C > 0, K > 1 and a sequence of (N + 1) × (N + 1) matrices A_N, satisfying the resolvent condition, such that ||(A_N|| ≥ C(N + 1)^1-∈ = C(n + 1)^1-∈ for N = n = 1, 2, 3, .... The result proved in this paper is also relevant to matrices A whose ∈-pseudospectra lie at a distance not exceeding K∈ from the unit disk for all ∈ > 0.