Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf¹ ; Michael Lin¹ ; Przemysław Wojtaszczyk²

¹ Ben-Gurion University of the Negev Beer Sheva, Israel
² Department of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Colloquium Mathematicum, Tome 124 (2011) no. 2, pp. 225-235

Résumé

Let $X$ be a Banach space with a basis. We prove that $X$ is reflexive if and only if every power-bounded linear operator $T$ satisfies Browder's equality $$ \Big\{ x\in X: \sup_n \Big\| \sum_{k=1}^n T^k x \Big\| \infty \Big\} = (I-T)X. $$ We then deduce that $X$ (with a basis) is reflexive if and only if every strongly continuous bounded semigroup $\{T_t: t\ge 0\}$ with generator $A$ satisfies $$ AX= \bigg\{x\in X:\sup_{s>0} \bigg\|\int_0^s\, T_tx\,dt \bigg\|\infty \bigg\}. $$ The range $(I-T)X$ (respectively, $AX$ for continuous time) is the space of $x \in X$ for which Poisson's equation $(I-T)y=x$ ($Ay=x$ in continuous time) has a solution $y \in X$; the above equalities for the ranges express sufficient (and obviously necessary) conditions for solvability of Poisson's equation.

Détail

DOI : 10.4064/cm124-2-7

Keywords: banach space basis prove reflexive only every power bounded linear operator satisfies browders equality sup sum x infty i t deduce basis reflexive only every strongly continuous bounded semigroup generator satisfies bigg sup bigg int bigg infty bigg range i t respectively continuous time space which poissons equation i t continuous time has solution above equalities ranges express sufficient obviously necessary conditions solvability poissons equation