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

摘要

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 greater than or equal to 1, the upper bounds parallel toA(n)parallel to less than or equal to eK(N + 1) and parallel toA(n)parallel to less than or equal to eK(n + 1). Here parallel to . parallel to is the spectral norm, K is the constant occurring in the resolvent condition, and the order of A is equal to N + 1 greater than or equal to 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 c > 0, there are fixed values C > 0, K > 1 and a sequence of (N + 1) x (N + 1) matrices AN, satisfying the resolvent condition, such that parallel to(A(N))(n)parallel to greater than or equal to C(N + 1)(1-epsilon) = C(n + 1)(1-epsilon) for N = n = 1, 2, 3,.... The result proved in this paper is also relevant to matrices A whose epsilon-pseudospectra lie at a distance not exceeding Kepsilon from the unit disk for all epsilon > 0. [References: 22]