Tauberian Operators (Operator Theory: Advances and Applications)

Tauberian operators were introduced to investigate a problem in summability theory from an abstract point of view. Since that introduction, they have made a deep impact on the isomorphic theory of Banach spaces. In fact, these operators havebeen useful in severalcontexts of Banachspacetheory that haveno apparent or obvious connections. For instance, they appear in the famous factorization of Davis, Figiel, Johnson and Pe lczynski [49] (henceforth the DFJP factorization), in the study of exact sequences of Banach spaces [174], in the solution of certain summabilityproblemsoftauberiantype[63,115], intheproblemoftheequivalence between the Krein-Milman property and the Radon-Nikodym property [151], in certain sequels of James characterization of re?exive Banach spaces [135], in the construction of hereditarily indecomposable Banach spaces [13], in the extension of the principle of local re?exivity to operators [27], in the study of certain Calkin algebras associated with the weakly compact operators [16], etc. Since the results concerning tauberian operatorsappear scattered throughout the literature, in this book wegive a uni?ed presentationof their propertiesand their main applications in functional analysis. We also describe some questions about tauberian operators that remain open. This book has six chapters and an appendix. In Chapter 1 we show how the concept of tauberian operator was introduced in the study of a classical problem in summability theory the characterization of conservative matrices that sum no bounded divergent sequences by means of functional analysis techniques. One of thosesolutionsisdue toCrawford[45], whoconsideredthe secondconjugateofthe operatorassociatedwithoneofthosematrices."

From the reviews:

“Tauberian operators were introduced by Kalton and Wilanski in 1976 as an abstract counterpart of some operators associated to conservative summability matrices. ... The book present in a clear and unified way the basic properties of tauberian operators and their applications in functional analysis scattered throughout the literature. ... is addressed to graduate students and researchers in functional analysis and operator theory, but it can be used also as a basic text for advanced graduate courses.” (V. Anisiu, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010)

“The monograph under review contains the first comprehensive exposition of properties and applications of Tauberian and co-Tauberian operators, as well as of those of operators belonging to various related classes. ... This monograph provides a careful unified account of ongoing research, and it is a welcome addition to the research literature on the qualitative theory of operators on Banach spaces. It is aimed at graduate students and researchers in operator theory and Banach spaces.” (Hans-Olav Tylli, Mathematical Reviews, Issue 2011 e)