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Spear Operators Between Banach Spaces

  • Book
  • © 2018

Overview

Authors:
  1. Vladimir Kadets

    1. School of Mathematics and Computer Science, V. N. Karazin Kharkiv National University, Kharkiv, Ukraine

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  2. Miguel Martín

    1. Departamento de Análisis Matemático, Universidad de Granada, Granada, Spain

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  3. Javier Merí

    1. Departamento de Análisis Matemático, Universidad de Granada, Granada, Spain

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  4. Antonio Pérez

    1. Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain

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  • No prerequisites required to fully understand an active research line
  • Full proofs of all the main results
  • Systematic study of spear operators for the first time in a book

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2205)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xvii
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  2. Historical Introduction: A Walk on the Results for Banach Spaces with Numerical Index 1

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages 1-36

  3. Spear Vectors and Spear Sets

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages 37-47

  4. Three Definitions for Operators: Spearness, the Alternative Daugavet Property, and Lushness

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages 49-66

  5. Some Examples in Classical Banach Spaces

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages 67-82

  6. Further Results

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages 83-95

  7. Isometric and Isomorphic Consequences

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages 97-102

  8. Lipschitz Spear Operators

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages 103-113

  9. Some Stability Results

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages 115-150

  10. Open Problems

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages 151-152

  11. Erratum to: Spear Operators Between Banach Spaces

    • Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
    Pages E1-E1

  12. Back Matter

    Pages 153-164
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Keywords

About this book

This monograph is devoted to the study of spear operators, that is, bounded linear operators G between Banach spaces X and Y satisfying that for every other bounded linear operator T:X → Y there exists a modulus-one scalar  ω such that

ǁ G+ωTǁ = 1 + ǁTǁ.


This concept extends the properties of the identity operator in those Banach spaces having numerical index one. Many examples among classical spaces are provided, being one of them the Fourier transform on L₁. The relationships with the Radon-Nikodým property, with Asplund spaces and with the duality, and some isometric and isomorphic consequences are provided. Finally, Lipschitz operators satisfying the Lipschitz version of the equation above are studied.

 
The book could be of interest to young researchers and specialists in functional analysis, in particular to those interested in Banach spaces and their geometry. It is essentially self-contained and only basic knowledge of functional analysis is needed.

Reviews

“This book will certainly be of interest to all researchers who specialise in Banach space theory.” (Jan-David Hardtke, zbMATH 1415.46002, 2019)

Authors and Affiliations

  • School of Mathematics and Computer Science, V. N. Karazin Kharkiv National University, Kharkiv, Ukraine

    Vladimir Kadets

  • Departamento de Análisis Matemático, Universidad de Granada, Granada, Spain

    Miguel Martín, Javier Merí

  • Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain

    Antonio Pérez

Bibliographic Information

  • Book Title: Spear Operators Between Banach Spaces

  • Authors: Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-71333-5

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing AG, part of Springer Nature 2018

  • Softcover ISBN: 978-3-319-71332-8Published: 17 April 2018

  • eBook ISBN: 978-3-319-71333-5Published: 16 April 2018

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XVII, 164

  • Number of Illustrations: 5 b/w illustrations

  • Topics: Analysis

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