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