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 obtain that X (with a basis) is reflexive if and only if every strongly continuous bounded semi-group {T t : t ≥ 0} with generator A satisfies The range (I − T )X (respectively, AX for continuous time) is the space of x ∈ X for which Poisson's equation (I − T )y = x (Ay = x in continuous time) has a solution y ∈ X; the above equalities for

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... he ranges express sufficent (and obviously necessary) conditions for solvability of Poisson's equation.