A resolvent estimate for operators with finite spectrum
Résumé
{\small Let $T$ be a}\textit{\small{} power bounded}{\small{} operator on a Banach space $E$ : $P(T)=\mbox{sup}{}_{k\geq0}\left\Vert T^{k}\right\Vert _{E\rightarrow E}<\infty$. The condition {}``$P(T)<\infty$'' is called }\textit{\small{} power boundedness condition. }{\small Then, setting $\rho(T)=\mbox{sup}{}_{\vert z\vert>1}\left(\vert z\vert-1\right)\left\Vert R(z,\, T)\right\Vert _{E\rightarrow E}$, where $R(z,\, T)$ stands for the resolvent of $T$ at point $z$, the inequality $\rho(T)\leq P(T)$ is obvious and the condition {}``$\rho(T)<\infty$ {}`` is known as the classical }\textit{\small resolvent condition.} \par\end{flushleft}{\small \par} \begin{flushleft} {\small Here, we suppose that $n\geq1$ is an integer and $T$ is an algebraic operator acting on a Banach space $E$, which minimal annihilating polynomial is of degree less or equal than $n.$ We also suppose that the spectrum $\sigma(T)$ of $T$ is included in the unit disc $\mathbb{D}=\left\{ z\in\mathbb{C}:\,\vert z\vert<1\right\} $. Then, $T$ is of course }\textit{\small power bounded. }{\small But what happens} {\small to the inequality $\rho(T)\leq P(T)$ ? We set $\rho_{\frac{1}{2}}(T)=\mbox{sup}{}_{\vert z\vert>1}\sqrt{\vert z\vert^{2}-1}\left\Vert R(z,\, T)\right\Vert _{E\rightarrow E}$, and show that under our hypothesis on $T$ :\[ \rho_{\frac{1}{2}}(T)\leq\delta\frac{n}{\sqrt{1-r}}P(T)\;\mbox{and}\;\overline{\mbox{lim}}{}_{n\rightarrow\infty}\frac{1}{n}\frac{\rho_{\frac{1}{2}}(T)}{P(T)}\leq\pi\sqrt{1+r},\] where $\delta=\pi\left(1+\sqrt{2}\right)+1,$ and $r=\mbox{max}{}_{\lambda\in\sigma(T)}\left|\lambda\right|\,.$ This result is a consequence of a Bernstein-type inequality for rational functions in $\mathbb{D}$ having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D}$, involving the Hardy norms $\left\Vert .\right\Vert _{H^{1}}$ and $\left\Vert .\right\Vert _{H^{2}}$. We show this Bernstein-type inequality and its asymptotic sharpness as $n\rightarrow\infty$ and $r\rightarrow1^{-}$.} \par\end{flushleft}{\small \par} \begin{flushleft} {\small We also show the asymptotic sharpness of our upper bound $\delta\frac{n}{\sqrt{1-r}}P(T)$ as $n\rightarrow\infty$ and $r\rightarrow1^{-}$ but this time in the following sense :}\textit{\small{} }{\small there exists a contraction $A_{r}$ on the Hilbert space $\left(\mathbb{C}^{n},\,\left|.\right|_{2}\right)$ of spectrum $\{r\}$ such that \[ \underline{\mbox{lim}}_{r\rightarrow1}(1-r)^{\frac{1}{2}-\beta}\rho_{\frac{1}{2}}\left(A_{r}\right)\geq\mbox{cot}\left(\frac{\pi}{4n}\right)=P(A_{r}).\mbox{cot}\left(\frac{\pi}{4n}\right),\] for all $\beta\in(0,\,\frac{1}{2})\,.$ We finally link our result to the well-known Kreiss Matrix Theorem which has been also proved using a Bernstein-type inequality for the same class of rational functions but involving this time the Hardy norms $\left\Vert .\right\Vert _{H^{1}}$ and $\left\Vert .\right\Vert _{H^{\infty}}\,.$} \par\end{flushleft}{\small \par}
Domaines
Analyse fonctionnelle [math.FA]
Origine : Fichiers produits par l'(les) auteur(s)