##
**Absolutely summing operators.**
*(English)*
Zbl 0855.47016

Cambridge Studies in Advanced Mathematics. 43. Cambridge: Cambridge Univ. Press. xv, 474 p. (1995).

The book by Diestel, Jarchow and Tonge gives a modern account of the theory of absolutely summing operators and its ramifications. These operators are basic for the understanding of absolute, unconditional and almost sure convergence of series in Banach spaces. The first six chapters and chapter 11 cover the fundamentals of \(p\)-summing operators and type/cotype in Banach spaces and can be used as a student text in lectures or seminars while other chapters treat more specialized topics or require more elaborate arguments or proofs. The book is vividly written and explains various aspects of the modern theory of Banach spaces to students or researchers working in other areas of mathematics. Each chapter contains a wealth of notes and remarks at the end giving further results, directions to be persued and bibliographical/historical information.

Chapter 1 covers the Dvoretzky-Rogers Theorem, i.e. the inequivalence of absolute and unconditional convergence of series in infinite-dimensional Banach spaces, and basic inequalities as Grothendieck’s and Khinchin’s inequality. The basic properties of \(p\)-summing operators are discussed in chapter 2. Lindenstrauss/Pelczynski’s work on these operators acting in \(L_p\)-spaces is the subject of chapter 3. Chapter 4 covers maps in Hilbert spaces. The \(p\)-integral and \(p\)-nuclear operators, close “relatives” of the \(p\)-summing maps, as well as the important concept of trace duality of operator ideals like those above, are the topics of chapters 5 and 6.

Factorization properties, e.g. operators factoring through Hilbert spaces or \(L_p\), are treated in chapters 7 and 9 which require ultraproduct techniques discussed in chapter 8. The important concepts of Banach spaces of type \(p\) and cotype \(q\) are studied in chapter 11, with applications to almost sure convergence of randomized series in chapter 12. The connection with \(p\)-convexity and \(q\)-concavity in Banach lattices is explained in chapter 16, chapter 17 covers the “local unconditional structure” of Banach spaces, a weaker property than the one of Banach lattices. Chapters 13 and 14 study spaces of finite cotype and give Pisier’s result of the equivalence of \(K\)-convexity and \(B\)-convexity of Banach spaces. Chapter 18 gives nice applications to Banach algebras and is of different nature than the other chapters. The last chapter 19 resumes the topic of the first chapter: as a strengthening of the Dvoretzky-Rogers lemma, the famous Theorem of Dvoretzky is proven, i.e. that every infinite-dimensional Banach space contains almost isometric Hilbertian sections of arbitrarily large finite dimension.

The book is a very useful addition to the existing literature on the geometry of Banach spaces.

Chapter 1 covers the Dvoretzky-Rogers Theorem, i.e. the inequivalence of absolute and unconditional convergence of series in infinite-dimensional Banach spaces, and basic inequalities as Grothendieck’s and Khinchin’s inequality. The basic properties of \(p\)-summing operators are discussed in chapter 2. Lindenstrauss/Pelczynski’s work on these operators acting in \(L_p\)-spaces is the subject of chapter 3. Chapter 4 covers maps in Hilbert spaces. The \(p\)-integral and \(p\)-nuclear operators, close “relatives” of the \(p\)-summing maps, as well as the important concept of trace duality of operator ideals like those above, are the topics of chapters 5 and 6.

Factorization properties, e.g. operators factoring through Hilbert spaces or \(L_p\), are treated in chapters 7 and 9 which require ultraproduct techniques discussed in chapter 8. The important concepts of Banach spaces of type \(p\) and cotype \(q\) are studied in chapter 11, with applications to almost sure convergence of randomized series in chapter 12. The connection with \(p\)-convexity and \(q\)-concavity in Banach lattices is explained in chapter 16, chapter 17 covers the “local unconditional structure” of Banach spaces, a weaker property than the one of Banach lattices. Chapters 13 and 14 study spaces of finite cotype and give Pisier’s result of the equivalence of \(K\)-convexity and \(B\)-convexity of Banach spaces. Chapter 18 gives nice applications to Banach algebras and is of different nature than the other chapters. The last chapter 19 resumes the topic of the first chapter: as a strengthening of the Dvoretzky-Rogers lemma, the famous Theorem of Dvoretzky is proven, i.e. that every infinite-dimensional Banach space contains almost isometric Hilbertian sections of arbitrarily large finite dimension.

The book is a very useful addition to the existing literature on the geometry of Banach spaces.

Reviewer: H.König (Kiel)

### MSC:

47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |

46B20 | Geometry and structure of normed linear spaces |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |