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 {x is an element of X : sup(n) parallel to Sigma(n)(k=1) T(k)x parallel to < infinity} = (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 >= 0} with generator A satisfies AX = {x is an element of X : sup(s>0)parallel to integral(a)(0) T(t)x dt parallel to < infinity}. The range (I - T)X (respectively, A X for continuous time) is the space of x is an element of X for which Poisson's equation (I - T)y = x (Ay = x in continuous time) has a solution y is an element of X; the above equalities for the ranges express sufficient (and obviously necessary) conditions for solvability of Poisson's equation.