Operators with small Kreiss constants

Nikolaos Chalmoukis; Georgios Tsikalas; Dmitry Yakubovich

Paper

Operators with small Kreiss constants

Abstract

We investigate matrices satisfying the Kreiss condition

with

lying arbitrarily close to

We provide lower bounds for the power growth of such matrices, which complement and refine related estimates due to Nikolski and Spijker-Tracogna-Welfert. We also study operators that satisfy a variant of the above Kreiss condition where

is replaced by

, where the positive continuous function

tends to

We show that, if the spectum of

touches the unit circle only at a single point and the resolvent of

satisfies a growth restriction along the unit circle, it is possible to choose

so that this Kreiss-type condition guarantees similarity to a contraction. At the core of our proof lies a positivity argument involving the double-layer potential operator. Counterexamples related to less restrictive choices of

are also provided.