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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 ≥ 1, the upper bounds A n ≤ eK(N + 1) and A n ≤ eK(n + 1). Here · 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 thatThe 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.

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