Martingales in Banach Spaces

This book focuses on applications of martingales to the geometry of Banach spaces, and is accessible to graduate students.

Gilles Pisier (Author)

9781107137240, Cambridge University Press

Hardback, published 6 June 2016

580 pages, 11 b/w illus.
23.5 x 15.7 x 3.6 cm, 0.93 kg

'This book is devoted to various aspects a?rming the importance of martingale techniques throughout the development of modern Banach space theory. … The book is self-contained and is quite accessible with only a basic functional analysis background. In particular, it does not assume any prior knowledge of scalar-valued martingale theory. … It is this reviewer's opinion that this excellent book will appeal to a wide audience and will become a classic reference in martingale theory.' Narcisse Randrianantoanina, Mathematical Reviews

This book focuses on the major applications of martingales to the geometry of Banach spaces, and a substantial discussion of harmonic analysis in Banach space valued Hardy spaces is also presented. It covers exciting links between super-reflexivity and some metric spaces related to computer science, as well as an outline of the recently developed theory of non-commutative martingales, which has natural connections with quantum physics and quantum information theory. Requiring few prerequisites and providing fully detailed proofs for the main results, this self-contained study is accessible to graduate students with a basic knowledge of real and complex analysis and functional analysis. Chapters can be read independently, with each building from the introductory notes, and the diversity of topics included also means this book can serve as the basis for a variety of graduate courses.

Introduction
Description of the contents
1. Banach space valued martingales
2. Radon Nikodým property
3. Harmonic functions and RNP
4. Analytic functions and ARNP
5. The UMD property for Banach spaces
6. Hilbert transform and UMD Banach spaces
7. Banach space valued H1 and BMO
8. Interpolation methods
9. The strong p-variation of martingales
10. Uniformly convex of martingales
11. Super-reflexivity
12. Interpolation and strong p-variation
13. Martingales and metric spaces
14. Martingales in non-commutative LP *.

Subject Areas: Mathematical physics [PHU], Stochastics [PBWL], Probability & statistics [PBT], Analytic geometry [PBMS], Geometry [PBM], Functional analysis & transforms [PBKF], Mathematics [PB]