Poisson's equation and characterizations of reflexivity of banach spaces

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 We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup {Tt: t ≥ 0} with generator A satisfies The range (I - T)X (respectively, AX for continuous time) is the space of x ε X for which Poisson's equation (I-T)y = x (Ay = x in continuous time) has a solution y ε X; the above equalities for the ranges express sufficient (and obviously necessary) conditions for solvability of Poisson's equation.