Tauberian operators (Q841999)

Originally introduced by \textit{N.\,Kalton} and \textit{A.\,Wilansky} in [Proc.\ Am.\ Math.\ Soc.\ 57, 251--255 (1976; Zbl 0304.47023)], a bounded linear operator \(T:X \to Y\) between Banach spaces \(X\) and \(Y\) is said to be tauberian whenever \({T^{\ast \ast}}^{-1}(Y) \subset X\) (where \(T^{\ast \ast}: X^{\ast \ast} \rightarrow Y^{\ast \ast}\) denotes the second conjugate of \(T\)). From the backcover: ``This book gives a complete exposition of the theory of tauberian operators from the very basic results to the most recent advances making emphasis in its applications to Banach space theory. After describing the origins of the subject in the study of summability of series, it presents the general theory for tauberian operators on abstract Banach spaces. The main sources of examples of tauberian operators are described in detail, including the case of operators on spaces of integrable functions.'' The above is a well-fitting self-description of the monograph under review, as well as the following excerpt from the preface: ``This book is addressed to graduate students and researchers interested in functional analysis and operator theory. The prerequisites for reading this book are a basic knowledge of functional analysis, including the consequences of the Hahn-Banach theorem and the open mapping theorem. Familiarity with the rudiments of Fredholm theory for operators and some parts of Banach space theory, like criteria for the existence of basic subsequences from a given sequence, Rosenthal's \(\ell_1\)-theorem, ultraproducts and the principle of local reflexivity would be helpful.'' The contents of the book are organized as follows: In Chapter~1, the authors show how the concept of tauberian operators was introduced in the study of a classical problem in summability theory by means of functional analysis techniques. Chapter~2 displays the basic structural properties of the class of tauberian operators, presents some basic examples and describes the most important characterizations of tauberian operators. Chapter~3 deals with duality theory and cotauberian operators as well as -- probably the heart of the theory -- an improved version of the DFJP factorization (named after an article by \textit{W.\,J.\,Davis, T.\,Figiel, W.\,B.\thinspace Johnson} and \textit{A.\,Pełczyński} [J.~Funct.\ Anal.\ 17, 311--327 (1974; Zbl 0306.46020)]), which allows to show plenty of examples of tauberian and cotauberian operators: every bounded linear operator \(T:X \rightarrow Y\) between Banach spaces can be factorized as \(T=jUk\), with \(j\) tauberian, \(k\) cotauberian and \(U\) a bijective isomorphism. In Chapter~5, the authors strongly focus on their own contributions to the study of tauberian operators \(T: L_1(\mu) \rightarrow Y\), where \(\mu\) is a finite measure and \(Y\) a Banach space. This is followed in Chapter~5 by an exposition of the main applications of tauberian operators in Banach space theory. In the concluding Chapter~6, some classes of operators -- so-called operator semigroups -- that have a similar behaviour to that of tauberian operators are considered. Each chapter ends with a section of Notes and Remarks where the authors include some comments, complementary results and bibliographical references. For the convenience of the reader, a brief exposition of some of the aforementioned prerequisites has been included in an appendix. The presentation is very clear and readable. While the authors' intention has been to present a self-contained exposition of the fundamental results of the subject, they admit that sometimes they only give a reference instead of a complete proof when describing the applications. This only occurs in the more advanced parts of the book, so it is appropriate to assume that readers who have come this far will have obtained the necessary prerequisites to be able to successfully comprehend the given references. Altogether, the authors keep their promises and deliver a very nice introduction to the theory of tauberian operaters, both for graduate students and experienced researchers.