Synopses & Reviews
We can best understand many fundamental processes in analysis by studying and comparing the summability of series in various modes of convergence. This text provides the reader with basic knowledge of real and functional analysis, with an account of p-summing and related operators. The account is panoramic, with detailed expositions of the core results and highly relevant applications to harmonic analysis, probability and measure theory, and operator theory. This is the first time that the subject and its applications have been presented in such complete detail in book form. Graduate students and researchers in real, complex and functional analysis, and probability theory will benefit from this text.
"Beautifully motivated exposition!" American Mathematical Monthly
"...a very welcome addition to a rather long list of recently published monographs concerning deep mathematics in Banach spaces from the last three decades--it is the first time that the important subject of 'summing operators' has been presented in almost complete detail in book form. Large parts of the book are within reach of graduate students, extensive 'notes and remarks' sections fill even sophisticated corners of the field with light--this book promises to be a future classic....this excellent book will have a wide audience which consists not only of interested students: there will be many mathematicians inside and outside of Banach spcae theory who will use it as a rich source for references and inspiration. Wonderful mathematics presented in striking fashion!" Andreas Defant, Mathematical Reviews
This text provides the beginning graduate student with an account of p-summing and related operators.
Table of Contents
Introduction; 1. Unconditioned and absolute summability in Banach spaces; 2. Fundamentals of p-summing operators; 3. Summing operators on Cp-spaces; 4. Operators on Hilbert spaces and summing operators; 5. p-Integral operators; 6. Trace duality; 7. 2-Factorable operators; 8. Ultraproducts and local reflexivity; 9. p-Factorable operators; 10. (q, p)-Summing operators; 11. Type and cotype: the basics; 12. Randomised series and almost summing operators; 13. K-Convexity and B-convexity; 14. Spaces with finite cotype; 15. Weakly compact operators on C(K)-spaces; 16. Type and cotype in Banach lattices; 17. Local unconditionality; 18. Summing algebras; 19. Dvoretzky's theorem and factorization of operators; References; Indexes.