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

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.