Positive Operators

by Aliprantis, Charalambos D.; Burkinshaw, Owen

• ISBN13: 9781402050077
• ISBN10: 1402050070
• Format: Hardcover
• Copyright: 2006-10-31
• Publisher: Springer-Nature New York Inc

Summary

Reprinted by popular demand, this monograph presents a comprehensive study of positive operators between Riesz spaces and Banach lattices. Since the first publication of this book, (Academic Press, 1985), the subject of positive operators and Riesz spaces has found many applications in several disciplines, including social sciences and engineering. It is well known that many linear operators between Banach spaces arising in classical analysis are in fact positive operators. Therefore we study here positive operators in the setting of Riesz spaces and Banach lattices and from both the algebraic and topological points of view. Special emphasis is given to the compactness properties of positive operators and their relations to the order structures of the spaces the operators are acting upon. In order to make the book as self-sufficient as possible, some basic results from the theory of Riesz spaces and Banach lattices are included with proofs where necessary. However, familiarity with the elementary concepts of real analysis and functional analysis is assumed. The book is divided into five chapters, each consisting of nineteen sections all ending with exercises designed to supplement and illustrate the material.

Table of Contents

Foreword	ix
Historical Foreword	xi
Preface	xiii
Acknowledgments	xv
List of Special Symbols	xvii
Chapter 1. The Order Structure of Positive Operators	1	(78)
§1.1. Basic Properties of Positive Operators	1	(22)
§1.2. Extensions of Positive Operators	23	(8)
§1.3.Order Projections	31	(14)
§1.4. Order Continuous Operators	45	(13)
§1.5. Positive Linear Functionals	58	(21)
Chapter 2. Components, Homomorphisms, and Orthomorphisms	79	(54)
§2.1. The Components of a Positive Operator	80	(13)
§2.2. Lattice Homomorphisms	93	(18)
§2.3. Orthomorphisms	111	(22)
Chapter 3. Topological Considerations	133	(48)
§3.1. Topological Vector Spaces	133	(23)
§3.2. Weak Topologies on Banach Spaces	156	(13)
§3.3. Locally Convex-Solid Riesz Spaces	169	(12)
Chapter 4. Banach Lattices	181	(92)
§4.1. Banach Lattices with Order Continuous Norms	181	(26)
§4.2. Weak Compactness in Banach Lattices	207	(14)
§4.3. Embedding Banach Spaces	221	(33)
§4.4. Banach Lattices of Operators	254	(19)
Chapter 5. Compactness Properties of Positive Operators	273	(86)
§5.1. Compact Operators	273	(17)
§5.2. Weakly Compact Operators	290	(26)
§5.3. L- and M-weakly Compact Operators	316	(24)
§5.4. Dunford–Pettis Operators	340	(19)
Bibliography	359	(10)
Monographs	369	(2)
Index	371