Eigenfunction Expansions, Operator Algebras and Riemannian Symmetric Spaces: 1st Edition (Hardback) book cover

Eigenfunction Expansions, Operator Algebras and Riemannian Symmetric Spaces

1st Edition

By Robert M Kauffman

Chapman and Hall/CRC

152 pages

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Hardback: 9780582276345
pub: 1996-09-25
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Description

This Research Note pays particular attention to studying the convergence of the expansion and to the case where D is a family of partial differential operators. All operators in the natural von Neumann algebraassociated with D, and also unbounded operators affiliated with this algebra, are expanded simultaneously in terms of generalized eigenprojections. These are operators which carry a natural space associated with D into its dual. The elements of the range of these eigenprojections are the eigenfunctions, which solve the appropriate eigenvalue equations by duality. The spectral measure is abstractly defined, but its absolute continuity with respect to Hausdorf measure on the joint spectrum is shown to occur when the eigenfunctions are very well-behaved. Uniqueness results are given showing that any two expansions arise from each other by a simple change of variable.

A considerable effort has been made to keep the book self-contained for readers with a background in functional analysis including a basic understanding of the theory of von Neumann algebras. More advanced topics in functional analysis, andan introduction to differential geometry and differential operator theory, mostly without proofs, are given in an extensive section on background material.

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"a concise monographic treatment like this has been lacking in the literature… the general theory developed in this book will be of benefit for any analyst."

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Table of Contents

1 Introduction

2 Some background material

3 Generalized eigenprojection expansions

4 Eigenfunction expansions on Cn

5 When A exists

6 Differential operators on Riemannian manifolds

7 Operator algebras and symmetric spaces

8 Spaces of noncompact type

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
MAT037000
MATHEMATICS / Functional Analysis