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2018 | Buch

Historical Introduction: A Walk on the Results for Banach Spaces with Numerical Index 1 Kapitel lesen Erstes Kapitel lesen

Spear Operators Between Banach Spaces

verfasst von: Prof. Dr. Vladimir Kadets, Prof. Dr. Miguel Martín, Prof. Javier Merí, Prof. Antonio Pérez

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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 $T:X\longrightarrow Y$ there exists a modulus-one scalar $\omega$ such that$\|G + \omega\,T\|=1+ \|T\|$.
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 $L_1$. 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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Historical Introduction: A Walk on the Results for Banach Spaces with Numerical Index 1
Abstract
This chapter contains an overview of the known results about Banach spaces with numerical index 1, as well as the notation and terminology we will need along the book.
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Chapter 2. Spear Vectors and Spear Sets
Abstract
We recall the concept of spear vector and introduce the new notion of spear set. They are both used as “leitmotiv” to give a unified presentation of the concepts of spear operator, lush operator, aDP, and other type of operators that will be introduced here. We collect some properties of spear sets and vectors, together with some (easy) examples of spear vectors.
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Chapter 3. Three Definitions for Operators: Spearness, the Alternative Daugavet Property, and Lushness
Abstract
This is the main chapter of our manuscript, as we introduce and deeply study the main definitions: the one of spear operator, the weaker of operator with the alternative Daugavet property, and the stronger of lush operator.
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Chapter 4. Some Examples in Classical Banach Spaces
Abstract
Our aim here is to present examples of operators which are lush, spear, or have the aDP, defined in some classical Banach spaces. One of the most intriguing examples is the Fourier transform on L1, which we prove that is lush. Next, we study a number of examples of operators arriving to spaces of continuous functions. In particular, it is shown that every uniform algebra is lush-embedded into a space of bounded continuous functions. Finally, examples of operators acting from spaces of integrable functions are studied.
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Chapter 5. Further Results
Abstract
Our goal here is to complement the previous chapter with some interesting results. We characterize lush operators when the domain space has the Radon-Nikodým Property or the codomain space is Asplund, and we get better results when the domain or the codomain is finite-dimensional or when the operator has rank one. Further, we study the behaviour of lushness, spearness and the aDP with respect to the operation of taking adjoint operators.
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Chapter 6. Isometric and Isomorphic Consequences
Abstract
Our goal here is to present consequences on the Banach spaces X and Y of the fact that there is \(G\in \mathcal {L}(X,Y)\) which is a spear operator, is lush, or has the aDP.
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Chapter 7. Lipschitz Spear Operators
Abstract
We study Lipschitz spear operators. These are just the spear vectors of the space of Lipschitz operators between two Banach spaces endowed with the Lipschitz norm. The main result here is that every (linear) lush operator is a Lipschitz spear operator. We also provide with analogous results for aDP operators and for Daugavet centers.
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Chapter 8. Some Stability Results
Abstract
Our aim here is to provide several results on the stability of our properties for operators by several operations like absolute sums, vector-valued function spaces, and ultraproducts.
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Chapter 9. Open Problems
Abstract
We complete the book with a collection of open problems.
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Erratum to: Spear Operators Between Banach Spaces
Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez
Backmatter
Metadaten
Titel
Spear Operators Between Banach Spaces
verfasst von
Prof. Dr. Vladimir Kadets
Prof. Dr. Miguel Martín
Prof. Javier Merí
Prof. Antonio Pérez
Copyright-Jahr
2018
Electronic ISBN
978-3-319-71333-5
Print ISBN
978-3-319-71332-8
DOI
https://doi.org/10.1007/978-3-319-71333-5