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La aproximación geométrico-secuencial en los problemas de optimización dinámicos. I. El principio de máximo puntual.

Miguel Martín Dávila — 1985

Trabajos de Estadística e Investigación Operativa

En este artículo introducimos una nueva metodología para la generación de condiciones necesarias en problemas de optimización dinámicos. Denominamos a esta metodología la aproximación secuencial en contraposición a la aproximación puntual clásica y mostramos cómo obtener un principio de máximo puntual con este método.

Convexity around the Unit of a Banach Algebra

Kadets, VladimirKatkova, OlgaMartín, MiguelVishnyakova, Anna — 2008

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary: 46B20. Secondary: 46H99, 47A12. We estimate the (midpoint) modulus of convexity at the unit 1 of a Banach algebra A showing that inf {max±||1 ± x|| − 1 : x ∈ A, ||x||=ε} ≥ (π/4e)ε²+o(ε²) as ε → 0. We also give a characterization of two-dimensional subspaces of Banach algebras containing the identity in terms of polynomial inequalities.

Extremely non-complex Banach spaces

Miguel MartínJavier Merí — 2011

Open Mathematics

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.

Numerical index of vector-valued function spaces

Miguel MartínRafael Payá — 2000

Studia Mathematica

We show that the numerical index of a c 0 -, l 1 -, or l -sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K,X) and L 1 ( μ , X ) (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.

The alternative Dunford-Pettis Property in the predual of a von Neumann algebra

Miguel MartínAntonio M. Peralta — 2001

Studia Mathematica

Let A be a type II von Neumann algebra with predual A⁎. We prove that A⁎ does not have the alternative Dunford-Pettis property introduced by W. Freedman [7], i.e., there is a sequence (φₙ) converging weakly to φ in A⁎ with ||φₙ|| = ||φ|| = 1 for all n ∈ ℕ and a weakly null sequence (xₙ) in A such that φₙ(xₙ) ↛ 0. This answers a question posed in [7].

The Daugavet equation for polynomials

Yun Sung ChoiDomingo GarcíaManuel MaestreMiguel Martín — 2007

Studia Mathematica

We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ||Id + P|| = 1 + ||P|| is satisfied for all weakly compact polynomials P: X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation m a x | ω | = 1 | | I d + ω P | | = 1 + | | P | | for polynomials P: X → X. We show that this equation holds for every polynomial on the complex space X =...

On the Bishop-Phelps-Bollobás theorem for operators and numerical radius

Sun Kwang KimHan Ju LeeMiguel Martín — 2016

Studia Mathematica

We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ₁-sums and -sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X ⊕₁ Y has the weak BPBp-nu, then (X,Y) has the Bishop-Phelps-Bollobás property (BPBp). On the other hand, if Y is strongly lush and X Y has the weak BPBp-nu, then (X,Y) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L₁(μ)...

On order structure and operators in L ∞(μ)

It is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.

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

Kostyantyn BoykoVladimir KadetsMiguel MartínJavier 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...

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