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Poisson's equation and characterizations of reflexivity of Banach spaces

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Fonf V., Lin M., Wojtaszczyk P.

Colloquium Mathematicum

2011

tom 124

nr 2

225-235

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.

On separable Banach spaces containing all separable reflexive Banach spaces

99%

Wojtaszczyk P.

Studia Mathematica

1970-1971

tom 37

nr 2

197-202

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

2 Wojtaszczyk P.

1 Fonf V.

1 Lin M.

1 2011

1 1971

Wyniki wyszukiwania

Poisson's equation and characterizations of reflexivity of Banach spaces

On separable Banach spaces containing all separable reflexive Banach spaces