An Improvement of the Resolvent Estimate in the Kreiss Matrix Theorem

TL;DR

This paper enhances the resolvent estimate in the Kreiss matrix theorem, providing a more accurate description of resolvent behavior at infinity for matrices generating bounded semigroups.

Contribution

It introduces an improved resolvent estimate that is equivalent to Kreiss's condition and better captures the behavior at infinity.

Findings

New resolvent estimate is equivalent to Kreiss's condition

Better describes resolvent behavior at infinity

Applicable to matrices generating bounded semigroups

Abstract

We improve the resolvent estimate in the Kreiss matrix theorem for a set of matrices that generate uniformly bounded semigroups. The new resolvent estimate is proved to be equivalent to Kreiss's resolvent condition, and it better describes the behavior of the resolvents at infinity.