Analysis in Banach Spaces: Volume II: Probabilistic Methods and Operator Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 67)

This second volume of Analysis in Banach Spaces, Probabilistic Methods and Operator Theory, is the successor to Volume I, Martingales and Littlewood-Paley Theory. It presents a thorough study of the fundamental randomisation techniques and the operator-theoretic aspects of the theory. The first two chapters address the relevant classical background from the theory of Banach spaces, including notions like type, cotype, K-convexity and contraction principles. In turn, the next two chapters provide a detailed treatment of the theory of R-boundedness and Banach space valued square functions developed over the last 20 years. In the last chapter, this content is applied to develop the holomorphic functional calculus of sectorial and bi-sectorial operators in Banach spaces. Given its breadth of coverage, this book will be an invaluable reference to graduate students and researchers interested in functional analysis, harmonic analysis, spectral theory, stochastic analysis, and the operator-theoretic approach to deterministic and stochastic evolution equations. "The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem. Over the past fifteen years, motivated by regularity problems in evolution equations, there has been tremendous progress in the analysis of Banach space-valued functions and processes. The contents of this extensive and powerful toolbox have been mostly scattered around in research papers and lecture notes. Collecting this diverse body of material into a unified and accessible presentation fills a gap in the existing literature. The principal audience that we have in mind consists of researchers who need and use Analysis in Banach Spaces as a tool for studying problems in partial differential equations, harmonic analysis, and stochastic analysis. Self-contained and offering complete proofs, this work is accessible to graduate students and researchers with a background in functional analysis or related areas."--Publisher's description for v. 1 Front Matter ....Pages i-xxiii Random sums (Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis)....Pages 1-52 Type, cotype, and related properties (Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis)....Pages 53-162 R-boundedness (Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis)....Pages 163-250 Square functions and radonifying operators (Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis)....Pages 251-358 The H∞-functional calculus (Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis)....Pages 359-478 Back Matter ....Pages 479-616