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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.
Mikhail Popov, Chernivtsi National University, Ukraine; Miami University, Oxford, USA; Beata Randrianantoanina, Miami University, Oxford, USA.
Chapter 1. Preliminaries
Chapter 2. Each small operator is narrow
Chapter 3. Applications to the geometry of Lp spaces for 0 < p < 1 5
Chapter 4. A very non-compact narrow operator
Chapter 5. Some deep results on narrow operators
Chapter 6. Weak embeddings of L1
Chapter 7. For what spaces X every operator T 2 L(Lp;X) is narrow?
Chapter 8. Ideal properties of narrow operators
Chapter 9. Daugavet type properties of Lorentz spaces with
Chapter 10. Narrow operators on vector lattices
Chapter 11. Some generalizations of narrow operators
Bibliography