Narrow Operators on Function Spaces and Vector Lattices

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.

"[...] This monograph presents many interesting and deep results revolving around the notion of a narrow operator. The theorems are well motivated and the proofs are very detailed. The authors have edited their text very carefully; also the clear typography makes for a good reading. There are hardly any typos [...] Experts in Banach space theory will appreciate having a copy of this volume at hand."
Dirk Werner, Zentralblatt für Mathematik

"The monograph presents a systematic exposition of the theory of narrow operators and their applications. The proofs are complete. The monograph can be helpful to all beginners as well as advanced researchers in Banach space theory." Mathematical Reviews