Poisson's equation and characterizations of reflexivity of Banach spaces
Volume 124 / 2011
Abstract
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.
