Sums of commuting operators with maximal regularity

Christian Le Merdy¹ ; Arnaud Simard¹

¹ Département de Mathématiques Université de Franche-Comté 25030 Besançon Cedex, France

Studia Mathematica, Tome 147 (2001) no. 2, pp. 103-118

Résumé

Let $Y$ be a Banach space and let $S\subset L_p$ be a subspace of an $L_p$ space, for some $p\in (1,\infty )$. We consider two operators $B$ and $C$ acting on $S$ and $Y$ respectively and satisfying the so-called maximal regularity property. Let ${\cal B}$ and ${\cal C}$ be their natural extensions to $S(Y)\subset L_p(Y)$. We investigate conditions that imply that ${\cal B} +{\cal C}$ is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if $Y$ is a UMD Banach lattice and $e^{-tB}$ is a positive contraction on $L_p$ for any $t\geq 0$.

Détail

DOI : 10.4064/sm147-2-1

Keywords: banach space subset subspace space infty consider operators acting respectively satisfying so called maximal regularity property cal cal their natural extensions subset investigate conditions imply cal cal closed has maximal regularity property extending theorems lamberton weis particular holds umd banach lattice tb positive contraction geq