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2017 | Buch

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Analysis in Banach Spaces

Volume II: Probabilistic Methods and Operator Theory

verfasst von: Prof. Dr. Tuomas Hytönen, Jan van Neerven, Prof. Mark Veraar, Prof. Dr. Lutz Weis

Verlag: Springer International Publishing

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Random sums

Abstract

This chapter provides a systematic investigation of vector-valued Gaussian and Rademacher random sums, which will be a key tool for subsequent developments throughout this volume. Here we concentrate on general properties of random sums that remain valid in arbitrary Banach spaces, while more specific results connected with the geometry of the underlying space are taken up in the following chapter. In the present chapter, we establish a range of comparison results relating the L^p norms of different types of random sums, as well as different L^p norms of a fixed sum. We also describe the dual and bi-dual of the spaces of random sequences, and characterise the convergence of infinite random series. In the final section, we compare the L^p norms of random sums and lacunary trigonometric sums.

Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis

Chapter 2. Type, cotype, and related properties

Abstract

In this chapter we connect some of the deeper properties of Rademacher sums and Gaussian sums to the geometry of the Banach space in which they live. We begin with a study of the notions of type and cotype, defined through nontrivial upper and lower bounds for random sums. Under the assumption of finite cotype, we prove a refined version of the contraction principle involving function (instead of constant) coefficients; for this we also develop a necessary minimum of the theory of summing operators. In the third section we prove geometric characterisations of type and cotype 2, due to Kwapień, and of non-trivial type and cotype, due to Maurey and Pisier. The fourth section is devoted to the notion of K-convexity and its connections with the duality of the random sequence spaces; this section culminates in Pisier's characterisation of K-convexity in terms of non-trivial type. The final section investigates the properties of multiple random sums involving products of Gaussian or Rademacher variables.

Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis

Chapter 3. R-boundedness

Abstract

This chapter provides a detailed study of the notion of (Rademacher) R-boundedness, its Gaussian analogue of γ-boundedness, and some relatives – essential tools in deeper manipulations of random sums arising in their applications to various domains of analysis. We discuss both the general operator-theoretic mechanisms of creating R-bounded families, and concrete sources and applications of R-boundedness in classical analysis. One section is dedicated to the central role of R-boundedness in the theory of Fourier multipliers, and another one to the R-boundedness of integral means and the range of sufficiently smooth operator-valued functions. In the final section, we characterise the situations in which R-boundedness coincides with other types of boundedness.

Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis

Chapter 4. Square functions and radonifying operators

Abstract

This chapter presents the theory of radonifying operators and explains their use as generalised square functions, which allows the extension of key ideas from classical Littlewood--Paley theory in L^p spaces to more general Banach spaces. The space of radonifying operators is shown to display several function-space-like properties, including Hölder-type duality, convergence and Fubini theorems. Pointwise multipliers of this space are characterised in terms of γ-boundedness. We also show that the generalised square function space admits canonical extensions of linear operators bounded on the scalar-valued L² space, avoiding the intrinsic difficulties of the L^p extension problem discussed in Volume I. Assuming type, cotype or related properties, we even obtain R-bounded extensions of bounded families of Hilbert space operators. In the final section we prove several embedding theorems of classical function spaces into the space of radonifying operators.

Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis

Chapter 5. The H ∞-functional calculus

Abstract

In this final chapter we present a substantial application of the tools developed in the earlier chapters by constructing and characterising the H^∞-functional calculus of sectorial and bi-sectorial operators in a Banach space. We begin with a general theory of the sectorial operators and a construction of the H^∞-calculus, complemented by several examples. In two sections, we connect the H^∞-calculus to R-boundedness on the one hand, and to the generalised square functions on the other hand. In an independent section, we show the necessity of certain assumptions on the underlying Banach space to some of the deeper results about the H^∞-calculus. In two final sections, we discuss the H^∞-calculus of bi-sectorial operators, and establish the H^∞-calculus of generators of appropriate groups and semigroups on UMD spaces.

Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis

Backmatter

Titel: Analysis in Banach Spaces
verfasst von: Prof. Dr. Tuomas Hytönen
Jan van Neerven
Prof. Mark Veraar
Prof. Dr. Lutz Weis
Copyright-Jahr: 2017
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-69808-3
Print ISBN: 978-3-319-69807-6
DOI: https://doi.org/10.1007/978-3-319-69808-3

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