Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf ; Michael Lin ; Przemysław Wojtaszczyk

Colloquium Mathematicae, Tome 122 (2011), p. 225-235 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $x ∈ X: sup_{n} ||∑_{k = 1}^{n} T^{k}x|| < ∞$ = (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 $AX = {x ∈ X: sup_{s>0} ||∫_{0}^{s} T_{t}xdt|| < ∞}$. 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 express sufficient (and obviously necessary) conditions for solvability of Poisson’s equation.

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Publié le : 2011-01-01

Zbl 1242.46016

EUDML-ID : urn:eudml:doc:283593