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Daugavet centers and direct sums of Banach spaces

Tetiana Bosenko (2010)

Open Mathematics

A linear continuous nonzero operator G: X → Y is a Daugavet center if every rank-1 operator T: X → Y satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X 1⊕F X 2 of two Banach spaces X 1 and X 2 by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X 1 and X 2 there exists a Daugavet center acting from X 1⊕F X 2, and the class of those F such that for some pair of spaces X 1 and X 2 there is a Daugavet center...

Decomposition of Banach Space into a Direct Sum of Separable and Reflexive Subspaces and Borel Maps

Plichko, Anatolij (1997)

Serdica Mathematical Journal

* This paper was supported in part by the Bulgarian Ministry of Education, Science and Technologies under contract MM-506/95.The main results of the paper are: Theorem 1. Let a Banach space E be decomposed into a direct sum of separable and reflexive subspaces. Then for every Hausdorff locally convex topological vector space Z and for every linear continuous bijective operator T : E → Z, the inverse T^(−1) is a Borel map. Theorem 2. Let us assume the continuum hypothesis. If a Banach space E cannot...

Decompositions for real Banach spaces with small spaces of operators

Manuel González, José M. Herrera (2007)

Studia Mathematica

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces X i for which ( X i ) / ℐ n ( X i ) is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces X i can be divided into subsets in such a way that if X i and X j are in different subsets,...

Deformation of Banach spaces

Józef Banaś, Krzysztof Fraczek (1993)

Commentationes Mathematicae Universitatis Carolinae

Using some moduli of convexity and smoothness we introduce a function which allows us to measure the deformation of Banach spaces. A few properties of this function are derived and its applicability in the geometric theory of Banach spaces is indicated.

Denseness of norm attaining mappings.

María D. Acosta (2006)

RACSAM

The Bishop-Phelps Theorem states that the set of (bounded and linear) functionals on a Banach space that attain their norms is dense in the dual. In the complex case, Lomonosov proved that there may be a closed, convex and bounded subset C of a Banach space such that the set of functionals whose maximum modulus is attained on C is not dense in the dual. This paper contains a survey of versions for operators, multilinear forms and polynomials of the Bishop-Phelps Theorem. Lindenstrauss provided examples...

Denting point in the space of operator-valued continuous maps.

Ryszard Grzaslewicz, Samir B. Hadid (1996)

Revista Matemática de la Universidad Complutense de Madrid

In a former paper we describe the geometric properties of the space of continuous functions with values in the space of operators acting on a Hilbert space. In particular we show that dent B(L(H)) = ext B(L(H)) if dim H < 8 and card K < 8 and dent B(L(H)) = 0 if dim H < 8 or card K = 8, and x-ext C(K,L(H)) = ext C(K,L(H)).

Descriptive compact spaces and renorming

L. Oncina, M. Raja (2004)

Studia Mathematica

We study the class of descriptive compact spaces, the Banach spaces generated by descriptive compact subsets and their relation to renorming problems.

Descriptive Sets and the Topology of Nonseparable Banach Spaces

Hansell, R. (2001)

Serdica Mathematical Journal

This paper was extensively circulated in manuscript form beginning in the Summer of 1989. It is being published here for the first time in its original form except for minor corrections, updated references and some concluding comments.

Diameter 2 properties and convexity

Trond Arnold Abrahamsen, Petr Hájek, Olav Nygaard, Jarno Talponen, Stanimir Troyanski (2016)

Studia Mathematica

We present an equivalent midpoint locally uniformly rotund (MLUR) renorming of C[0,1] with the diameter 2 property (D2P), i.e. every non-empty relatively weakly open subset of the unit ball has diameter 2. An example of an MLUR space with the D2P and with convex combinations of slices of arbitrarily small diameter is also given.

Diameter, extreme points and topology

J. C. Navarro-Pascual, M. G. Sanchez-Lirola (2009)

Studia Mathematica

We study the extremal structure of Banach spaces of continuous functions with the diameter norm.

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