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."
