Operators with small Kreiss constants

TL;DR

This paper studies matrices with near-optimal Kreiss constants, providing bounds on their power growth, and explores conditions under which such operators are similar to contractions, using positivity and potential theory.

Contribution

It offers new lower bounds for matrices with small Kreiss constants and characterizes when these operators are similar to contractions under a refined Kreiss condition.

Findings

Lower bounds for power growth of matrices with small Kreiss constants

Conditions for similarity to contractions based on Kreiss-type conditions

Counterexamples for less restrictive Kreiss conditions

Abstract

We investigate matrices satisfying the Kreiss condition $$\|(zI-T)^{-1}\|\le\cfrac{K}{|z|-1}, \hspace{0.7 cm} |z|>1, $$ with $K$ lying arbitrarily close to $1.$ 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 $K$ is replaced by $1+\epsilon(|z|)$, where the positive continuous function $\epsilon(|z|)$ tends to $0$ as $|z|\to 1^+.$ We show that, if the spectrum of $T$ touches the unit circle only at a single point and the resolvent of $T$ satisfies a growth restriction along the unit circle, it is possible to choose $\epsilon$ 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 $\epsilon$ are also provided.