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Pseudodifferential Analysis on Symmetric Cones : Studies in Advanced Mathematics - Andre Unterberger

Pseudodifferential Analysis on Symmetric Cones

Studies in Advanced Mathematics

Hardcover Published: 13th December 1995
ISBN: 9780849378737
Number Of Pages: 224

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Symmetric cones, possibly disguised under non-linear changes of coordinates, are the building blocks of manifolds with edges, corners, or conical points of a very general nature. Besides being a canonical open set of some Euclidean space, a symmetric cone L has an intrinsic Riemannian structure of its own, turning it into a symmetric space. These two structures make it possible to define on L a pseudodifferential analysis (the Fuchs calculus). The considerable interest in pseudodifferential problems on manifolds with non-smooth boundaries makes the precise analyses presented in this book both interesting and important. Much of the material in this book has never been previously published.
The methods used throughout the text rely heavily on the use of tools from quantum mechanics, such as representation theory and coherent states. Classes of operators defined by their symbols are given intrinsic characterizations. Harmonic analysis is discussed via the automorphism group of the complex tube over L.
The basic definitions governing the Fuchs calculus are provided, and a thorough exposition of the fundamental facts concerning the geometry of symmetric cones is given. The relationship with Jordan algebras is outlined and the general theory is illustrated by numerous examples.
The book offers the reader the technical tools for proving the main properties of the Fuchs calculus, with an emphasis on using the non-Euclidean Riemannian structure of the underlying cone. The fundamental results of pseudodifferential analysis are presented. The authors also develop the relationship to complex analysis and group representation.
This book benefits researchers interested in analysis on non-smooth domains or anyone working in pseudodifferential analysis. People interested in the geometry or harmonic analysis of symmetric cones will find in this valuable reference a new range of applications of complex analysis on tube-type symmetric domains and of the theory of Jordan algebras.

Introductionp. 1
Pseudodifferential Analysis on R[superscript n]p. 11
General Definition of the Fuchs Calculusp. 29
The Geometry of Symmetric Conesp. 35
The Covariance Group of the Fuchs Calculusp. 59
Geometric Inequalitiesp. 67
Geometric Differential Inequalitiesp. 81
Weights and Classes of Symbolsp. 87
The Family of [mu]-Symbolsp. 93
Coherent Statesp. 101
From Symbols to Operators: The Main Estimate and Continuityp. 109
From Operators to Symbols: The Converse of the Main Estimatep. 115
Asymptotic Expansionsp. 123
A Beals-type Characterization of Operators of Classical Typep. 135
Action of Diffeomorphisms on Operators of Classical Typep. 149
The [lambda]-Weyl Calculus: Unbounded Realizationp. 161
Contraction of the [lambda]-Weyl Calculusp. 169
The [lambda]-Weyl Calculus: Bounded Realizationp. 181
Contraction of the [lambda]-Weyl Calculus (Bounded Realization)p. 191
Referencesp. 207
Index of Notationsp. 213
Indexp. 216
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780849378737
ISBN-10: 0849378737
Series: Studies in Advanced Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 224
Published: 13th December 1995
Publisher: CRC PR INC
Country of Publication: US
Dimensions (cm): 24.77 x 16.51  x 1.91
Weight (kg): 0.47
Edition Number: 1

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