Advanced Search

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 $x\in X:su{p}_n\left|\right|{\sum }_{k=1}^n{T}^{k}x\left|\right|<\infty$ = (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 $AX=x\in X:su{p}_{s>0}\left|\right|{\int }_0^s{T}_{t}xdt\left|\right|<\infty$. 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 the ranges...