A survey on the numerical index of a Banach space.
Miguel Martín (2000)
Extracta Mathematicae
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Miguel Martín (2000)
Extracta Mathematicae
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Karahanyan, M.I. (2005)
Lobachevskii Journal of Mathematics
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Gaur, A.K., Husain, T. (1989)
International Journal of Mathematics and Mathematical Sciences
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María D. Acosta, M. Ruiz Galán (2000)
Extracta Mathematicae
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In this note we deal with a version of James' Theorem for numerical radius, which was already considered in [4]. First of all, let us recall that this well known classical result states that a Banach space satisfying that all the (bounded and linear) functionals attain the norm, has to be reflexive [16].
Miguel Martín, Rafael Payá (2000)
Studia Mathematica
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We show that the numerical index of a -, -, or -sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K,X) and (K any compact Hausdorff space, μ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.
Maria D. Acosta, Francisco J. Aguirre, Rafael Payá (1992)
Acta Universitatis Carolinae. Mathematica et Physica
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Kostyantyn Boyko, Vladimir Kadets, Miguel Martín, Javier Merí (2009)
Studia Mathematica
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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...
María D. Acosta (1990)
Extracta Mathematicae
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Miguel Martín, Javier Merí (2011)
Open Mathematics
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A Banach space X is said to be an extremely non-complex space if the norm equality ∥Id +T 2∥ = 1+∥T 2∥ holds for every bounded linear operator T on X. We show that every extremely non-complex Banach space has positive numerical index, it does not have an unconditional basis and that the infimum of diameters of the slices of its unit ball is positive.
María D. Acosta, Rafael Payá (1989)
Revista Matemática de la Universidad Complutense de Madrid
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In this note we discuss some results on numerical radius attaining operators paralleling earlier results on norm attaining operators. For arbitrary Banach spaces X and Y, the set of (bounded, linear) operators from X to Y whose adjoints attain their norms is norm-dense in the space of all operators. This theorem, due to W. Zizler, improves an earlier result by J. Lindenstrauss on the denseness of operators whose second adjoints attain their norms, and is also related to a recent result...