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

Rachid Zarouf (1)

1 Mathématiques Fondamentales

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}=\left\{ z\in\mathbb{C}:\,\vert z\vert<1\right\} and if C=sup_{k\geq0}\left\Vert T^{k}\right\Vert _{E\rightarrow E}, where E stands for \mathbb{C}^{n} endowed with a certain norm \left|.\right| , then \left\Vert R(1,\, T)\right\Vert _{E\rightarrow E}\leq C\left(3n/dist(1,\,\sigma(T))\right)^{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), \left\Vert R(\lambda,\, T)\right\Vert \leq C\left(5\pi/3+2\sqrt{2}\right)n^{3/2}/dist\left(\lambda,\,\sigma\right), for all \lambda\notin\sigma(T) such that \vert\lambda\vert\geq1.

Domaines: Analyse fonctionnelle [math.FA]

Rachid Zarouf. Sharpening a result by E.B. Davies and B. Simon. Comptes Rendus. Mathématique, 2009, Ser. I 347 (2009) ., pp.939--942. ⟨hal-00367564v2⟩