The sharpness of stability estimates corresponding to a general resolvent condition (Q1567552)

In a previous article published by the authors, upper bounds were given for the norm of the \(n\)-th power of matrices \(B\in\mathbb{C}^{s\times s}\) under the following generalized version of the Kreiss resolvent condition: \[ r (B) \leq 1,\;\bigl\|(\zeta I-B)^{-1}\bigr\|\leq{L\over \bigl(|\zeta |-1 \bigr)^k|\zeta-1|^l },\;1<|\zeta |<\rho, \] where \(L\) is a positive constant, \(\rho\) is a constant with \(1<\rho \leq\infty\) and \(k,l\) are fixed integers satisfying \(k\geq 0\), \(l\geq 0\), \(k+l>1\) (here \(r(B)\) and \(\|B \|\) denote respectively the spectral radius and the operator norm of the matrix \(B)\). In the present paper the sharpness of these upper bounds is examined. The main result (Theorem 2.1) shows that these upper bounds are the best possible for \((k,l)\neq(0,1)\) in the sense that no general bounds exist which grow more slowly, as \(n\to\infty\) and \(s\to\infty\), than the given bounds.