Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf, Michael Lin, and Przemysław Wojtaszczyk. "Poisson's equation and characterizations of reflexivity of Banach spaces." Colloquium Mathematicae 124.2 (2011): 225-235. <http://eudml.org/doc/283593>.

@article{VladimirP2011,
abstract = {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.},
author = {Vladimir P. Fonf, Michael Lin, Przemysław Wojtaszczyk},
journal = {Colloquium Mathematicae},
keywords = {reflexive Banach spaces; power-bounded operator; mean ergodic operator},
language = {eng},
number = {2},
pages = {225-235},
title = {Poisson's equation and characterizations of reflexivity of Banach spaces},
url = {http://eudml.org/doc/283593},
volume = {124},
year = {2011},
}

TY - JOUR
AU - Vladimir P. Fonf
AU - Michael Lin
AU - Przemysław Wojtaszczyk
TI - Poisson's equation and characterizations of reflexivity of Banach spaces
JO - Colloquium Mathematicae
PY - 2011
VL - 124
IS - 2
SP - 225
EP - 235
AB - 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.
LA - eng
KW - reflexive Banach spaces; power-bounded operator; mean ergodic operator
UR - http://eudml.org/doc/283593
ER -