Narrow operators on function spaces and vector lattices / Mikhail Popov, Beata Randrianantoanina.

Format:

Book

Author/Creator:

Popov, Mykhaĭlo Mykhaĭlovych.

Contributor:

Randrianantoanina, Beata.

Series:

De Gruyter Studies in Mathematics

De Gruyter studies in mathematics, 0179-0986 ; 45

De Gruyter Studies in Mathematics ; 45

Language:

English

Subjects (All):

Narrow operators.

Riesz spaces.

Function spaces.

Physical Description:

1 online resource (336 p.)

Edition:

1st ed.

Place of Publication:

Berlin : De Gruyter, 2013.

Language Note:

English

Summary:

Most classes of operators that are not isomorphic embeddings are characterized by some kind of a "smallness" condition. Narrow operators are those operators defined on function spaces that are "small" at {-1,0,1}-valued functions, e.g. compact operators are narrow. The original motivation to consider such operators came from theory of embeddings of Banach spaces, but since then they were also applied to the study of the Daugavet property and to other geometrical problems of functional analysis. The question of when a sum of two narrow operators is narrow, has led to deep developments of the theory of narrow operators, including an extension of the notion to vector lattices and investigations of connections to regular operators. Narrow operators were a subject of numerous investigations during the last 30 years. This monograph provides a comprehensive presentation putting them in context of modern theory. It gives an in depth systematic exposition of concepts related to and influenced by narrow operators, starting from basic results and building up to most recent developments. The authors include a complete bibliography and many attractive open problems.

Contents:

Frontmatter

Preface

Contents

Chapter 1. Introduction and preliminaries

Chapter 2. Each "small" operator is narrow

Chapter 3. Some properties of narrow operators with applications to nonlocally convex spaces

Chapter 4. Noncompact narrow operators

Chapter 5. Ideal properties, conjugates, spectrum and numerical radii of narrow operators

Chapter 6. Daugavet-type properties of Lebesgue and Lorentz spaces

Chapter 7. Strict singularity versus narrowness

Chapter 8. Weak embeddings of L1

Chapter 9. Spaces X for which every operator T ∈ ℒ (Lp;X) is narrow

Chapter 10. Narrow operators on vector lattices

Chapter 11. Some variants of the notion of narrow operators

Chapter 12. Open problems

Bibliography

Index of names

Subject index

Notes:

Description based upon print version of record.

Includes bibliographical references and indexes.

ISBN:

9783110263343

3110263343

OCLC:

826479699