Sharpening a result by E.B. Davies and B. Simon

TL;DR

This paper improves a bound on the resolvent norm of matrices with spectrum inside the unit disk, providing a sharper inequality that applies to all points outside the spectrum with magnitude at least one.

Contribution

The authors refine previous bounds on the resolvent norm for matrices with spectrum in the unit disk, extending the applicability and tightening the inequality.

Findings

Improved resolvent norm inequality for matrices with spectrum in the unit disk.

The new bound is sharper and more general than previous results.

The constant factor in the inequality is explicitly increased to 5π/3+2√2.

Abstract

E. B. Davies et B. Simon have shown (among other things) the following result: if T is an n\times n matrix such that its spectrum \sigma(T) is included in the open unit disc \mathbb{D}=\{z\in\mathbb{C}:\,|z|<1\} and if C=sup_{k\geq0}||T^{k}||_{E\rightarrow E}, where E stands for \mathbb{C}^{n} endowed with a certain norm |.|, then ||R(1,\, T)||_{E\rightarrow E}\leq C(3n/dist(1,\,\sigma(T)))^{3/2} where R(\lambda,\, T) stands for the resolvent of T at point \lambda. Here, we improve this inequality showing that under the same hypotheses (on the matrix T), ||R(\lambda,\, T)|| \leq C(5\pi/3+2\sqrt{2})n^{3/2}/dist(\lambda,\,\sigma), for all \lambda\notin\sigma(T) such that |\lambda|\geq1.