Optimal Domain and Integral Extension of Operators

by Okada, Susumu; Ricker, Werner; Perez, Enrique A. Sanchez

• ISBN13: 9783764386474
• ISBN10: 3764386479
• eBook ISBN(s): 9783764386481
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-07-04
• Publisher: Birkhauser

Summary

"This book treats the analysis of linear operators from a quasi-Banach function space into a Banach space. The idea is to extend the operator to its optimal domain space and to take advantage of this in analysing the original operator. Of particular interest are ideal properties such as compactness, weak compactness and complete continuity." "Since the book is directed not only at experienced researchers but also advanced research students, complete proofs are included and examples are worked out in detail. A good working knowledge of functional analysis, measure theory, operator theory and commutative harmonic analysis is assumed, whereas any new material is developed in a self-contained manners."--BOOK JACKET.

Table of Contents

Preface	p. ix
Introduction	p. 1
Quasi-Banach Function Spaces
General theory	p. 18
The p-th power of a quasi-Banach function space	p. 38
Completeness criteria	p. 50
Completely continuous operators	p. 55
Convexity and concavity properties of linear operators	p. 62
Vector Measures and Integration Operators
Vector measures	p. 104
Bochner and Pettis integrals	p. 148
Compactness properties of integration operators	p. 152
Concavity of L[superscript 1](v) and the integration operator L[subscript v] for a vector measure v	p. 168
Optimal Domains and Integral Extensions
Set functions associated with linear operators on function spaces	p. 182
Optimal domains	p. 185
Kernel operators	p. 199
p-th Power Factorable Operators
p-th power factorable operators	p. 210
Connections with L[superscript p](m[subscript T])	p. 214
Optimality	p. 222
Compactness criteria	p. 229
Factorization of p-th Power Factorable Operators through L[superscript q]-spaces
Basic results	p. 239
Generalized Maurey-Rosenthal factorization theorems	p. 248
Bidual (p, q)-power-concave operators	p. 267
Factorization of the integration operator	p. 283
Operators from Classical Harmonic Analysis
The Fourier transform	p. 298
Convolution operators acting in L[superscript 1] (G)	p. 319
Operators acting in L[superscript p](G) via convolution with functions	p. 335
Operators acting in L[superscript p](G) via convolution with measures	p. 347
p-th power factorability	p. 367
Bibliography	p. 379
List of Symbols	p. 389
Index	p. 395
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