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• # Artykuł - szczegóły

## Colloquium Mathematicum

2011 | 124 | 2 | 225-235

## Poisson's equation and characterizations of reflexivity of Banach spaces

EN

### Abstrakty

EN
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.

225-235

wydano
2011

### Twórcy

autor
• Ben-Gurion University of the Negev, Beer Sheva, Israel
autor
• Ben-Gurion University of the Negev, Beer Sheva, Israel
• Department of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland