On the condition number of a Kreiss matrix

In 2005, N. Nikolski proved among other things that for any r is an element of(0, 1) and any K >= 1, the condition number CN(T) = parallel to T parallel to center dot parallel to T-1 parallel to of any invertible n-dimensional complex Banach space operators T satisfying the Kreiss condition, with spectrum contained in {r = | z| < 1}, satisfies the inequality CN(T) <= CK(T) parallel to T parallel to n/r(n) where K( T) denotes the Kreiss constant of T and C > 0 is an absolute constant. He also proved that for r << 1/n, the latter bound is asymptotically sharp as n.8. In this note, we prove that this bound is actually achieved by a family of explicit n x n Toeplitz matrices with arbitrary singleton spectrum {lambda}subset of D/{0} and uniformly bounded Kreiss constant. Independently, we exhibit a sequence of Jordan blocks with Kreiss constants tending to 8 showing that Nikolski's inequality is still asymptotically sharp as K and n go to8.