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

• Autorzy: Kostyantyn Boyko , Vladimir Kadets , Miguel Martín , Javier Merí
• Języki publikacji: EN

Abstrakty

EN

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 determined, is stable under ultraproducts, and we characterize those spaces of the form X = (ℝⁿ,||·||) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46B04: Isometric theory of Banach spaces
• 47A12: Numerical range, numerical radius
• 46B03: Isomorphic theory (including renorming) of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2009
• Tom: 190
• Numer: 2
• Strony: 117-133

Daty

wydano

2009

Twórcy

autor

Kostyantyn Boyko

• Department of Mechanics and Mathematics, Kharkov National University, Pl. Svobody 4, 61077 Kharkov, Ukraine

autor

Vladimir Kadets

• Department of Mechanics and Mathematics, Kharkov National University, Pl. Svobody 4, 61077 Kharkov, Ukraine

autor

Miguel Martín

• Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

autor

Javier Merí

• Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Identyfikatory

DOI

10.4064/sm190-2-2