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Projetive generators and resolutions of identity in Banach spaces.

J. Orihuela, M. Valdivia (1989)

Revista Matemática de la Universidad Complutense de Madrid

We introduce the notion of projective generator on a given Banach space. Weakly countably determined and dual spaces with the Radon Nikodým property have projective generators. If a Banach space has projective generator, then it admits a projective resolution of the identity. When a Banach space and its dual both have a projective generator then the space admits a shrinking resolution of the identity. These results include previous ones of Amir and Lindenstrauss, John and Zizler, Gul?ko, Vaak, Tacon,...

Proper subspaces and compatibility

Esteban Andruchow, Eduardo Chiumiento, María Eugenia Di Iorio y Lucero (2015)

Studia Mathematica

Let 𝓔 be a Banach space contained in a Hilbert space 𝓛. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambickiĭ, we say that a bounded operator on 𝓔 is a proper operator if it admits an adjoint with respect to the inner product of 𝓛. A proper operator which is self-adjoint with respect to the inner product of 𝓛 is called symmetrizable. By a proper subspace 𝓢 we mean a closed subspace of 𝓔 which is the range of a proper projection. Furthermore,...

Proper uniform algebras are flat

R. C. Smith (2009)

Czechoslovak Mathematical Journal

In this brief note, we see that if A is a proper uniform algebra on a compact Hausdorff space X , then A is flat.

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 of lush spaces and applications to Banach spaces with numerical index 1

Kostyantyn Boyko, Vladimir Kadets, Miguel Martín, Javier Merí (2009)

Studia Mathematica

The concept of lushness, introduced recently, is a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to having numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c₀ can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably...

Property ( 𝐰𝐋 ) and the reciprocal Dunford-Pettis property in projective tensor products

Ioana Ghenciu (2015)

Commentationes Mathematicae Universitatis Carolinae

A Banach space X has the reciprocal Dunford-Pettis property ( R D P P ) if every completely continuous operator T from X to any Banach space Y is weakly compact. A Banach space X has the R D P P (resp. property ( w L ) ) if every L -subset of X * is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product X ⊗ π Y has property ( w L ) when X has the R D P P , Y has property ( w L ) , and L ( X , Y * ) = K ( X , Y * ) .

Property (wM*) and the unconditional metric compact approximation property

Ăsvald Lima (1995)

Studia Mathematica

The main objective of this paper is to give a simple proof for a larger class of spaces of the following theorem of Kalton and Werner. (a) X has property (M*), and (b) X has the metric compact approximation property Our main tool is a new property (wM*) which we show to be closely related to the unconditional metric approximation property.

Proximal normal structure and relatively nonexpansive mappings

A. Anthony Eldred, W. A. Kirk, P. Veeramani (2005)

Studia Mathematica

The notion of proximal normal structure is introduced and used to study mappings that are "relatively nonexpansive" in the sense that they are defined on the union of two subsets A and B of a Banach space X and satisfy ∥ Tx-Ty∥ ≤ ∥ x-y∥ for all x ∈ A, y ∈ B. It is shown that if A and B are weakly compact and convex, and if the pair (A,B) has proximal normal structure, then a relatively nonexpansive mapping T: A ∪ B → A ∪ B satisfying (i) T(A) ⊆ B and T(B) ⊆ A, has a proximal point in the sense that...

Pseudocomplémentation dans les espaces de Banach

Patric Rauch (1991)

Studia Mathematica

This paper introduces the following definition: a closed subspace Z of a Banach space E is pseudocomplemented in E if for every linear continuous operator u from Z to Z there is a linear continuous extension ū of u from E to E. For instance, every subspace complemented in E is pseudocomplemented in E. First, the pseudocomplemented hilbertian subspaces of L ¹ are characterized and, in L p with p in [1, + ∞[, classes of closed subspaces in which the notions of complementation and pseudocomplementation...

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