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Narrow operators on function spaces and vector lattices

Author: Mykhaĭlo Mykhaĭlovych Popov; Beata Randrianantoanina
Publisher: Berlin : De Gruyter, [2013]
Series: De Gruyter studies in mathematics, 45.
Edition/Format:   Book : EnglishView all editions and formats
Database:WorldCat
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  Read more...
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Details

Document Type: Book
All Authors / Contributors: Mykhaĭlo Mykhaĭlovych Popov; Beata Randrianantoanina
ISBN: 9783110263039 3110263033 9783119163798 ) 3119163791 (set (hardcover + e-book)
OCLC Number: 818735918
Description: xiii, 319 pages ; 25 cm.
Contents: Introduction and preliminaries --
Each "small" operator is narrow --
Applications to nonlocally convex spaces --
Noncompact narrow operators --
Ideal properties, conjugates, spectrum and numerical radii --
Daugavet-type properties of Lebesgue and Lorentz spaces --
Strict singularity versus narrowness --
Weak embeddings of L₁ --
Spaces X for which every operator T € L(Lp,X) is narrow --
Narrow operators on vector lattices --
Some variants of the notion of narrow operators --
Open problems.
Series Title: De Gruyter studies in mathematics, 45.
Responsibility: by Mikhail Popov, Beata Randrianantoanina.

Abstract:

"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."--Publisher's website.
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