On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem. (English) Zbl 0736.15015

Let \(A\) be an \(N\times N\) complex matrix. The condition (1) \(\| A^n\|\leq C_0\) for \(n=1,2,3,\ldots\), for a positive constant \(C_0\) implies the condition (2) \(zI-A\) is regular with \(\|(zI-A)^{- 1}\|\leq C_1(| z|-1)^{-1}\) for all complex \(z\) with \(| z|>1\).
H.-O. Kreiss BIT, Nord. Tidskr. Inf.-behandl. 2, 153–181 (1962; Zbl 0109.34702)] proved that (2) implies (1) with \(C_0\) depending on \(C_1\) and \(N\) only. R. J. LeVeque and L. N. Trefethen proved [BIT 24, 584–591 (1984; Zbl 0559.15018)] that (2) implies (1) if \(C_0=2eNC_1\) and they conjectured that the implication holds for the optimal value \(C_0=eNC_1\).
The author proves this conjecture, his proof relying on a lemma which gives an upper bound for the arc-length of the image of the unit circle in the complex plane under a rational function.


15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A45 Miscellaneous inequalities involving matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
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