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Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf, Michael Lin, Przemysław Wojtaszczyk (2011)

Colloquium Mathematicae

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 : s u p 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 A X = x ∈ X : s u p s > 0 | | ∫ 0 s T t x d t | | < ∞ . 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...

Preduals of Sobolev-Campanato spaces

Konrad Gröger, Lutz Recke (2001)

Mathematica Bohemica

We present definitions of Banach spaces predual to Campanato spaces and Sobolev-Campanato spaces, respectively, and we announce some results on embeddings and isomorphisms between these spaces. Detailed proofs will appear in our paper in Math. Nachr.

Product spaces generated by bilinear maps and duality

Enrique A. Sánchez Pérez (2015)

Czechoslovak Mathematical Journal

In this paper we analyse a definition of a product of Banach spaces that is naturally associated by duality with a space of operators that can be considered as a generalization of the notion of space of multiplication operators. This dual relation allows to understand several constructions coming from different fields of functional analysis that can be seen as instances of the abstract one when a particular product is considered. Some relevant examples and applications are shown, regarding pointwise...

Properly semi-L-embedded complex spaces

Angel Rodríguez Palacios (1993)

Studia Mathematica

We prove the existence of complex Banach spaces X such that every element F in the bidual X** of X has a unique best approximation π(F) in X, the equality ∥F∥ = ∥π (F)∥ + ∥F - π (F)∥ holds for all F in X**, but the mapping π is not linear.

Properties and applications of Tauberian operators.

Manuel González (1990)

Extracta Mathematicae

Tauberian operators, which appeared in response to a problem in summability [GaW, KW] have found application in several situations: factorization of operators [DFJP], preservation of isomorphic properties of Banach spaces [N, NR], equivalence between the Radon-Nikodym property and the Krein-Milman property [Sch], and generalized Fredholm operators [Ta, Y].This paper is a survey of the main properties and applications of Tauberian operators.

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