Positive Operators

388 Pages·2006·2.126 MB·English

by  Charalambos D. Aliprantis, Owen Burkinshaw

POSITIVE OPERATORS
Positive Operators
by
CHARALAMBOS D. ALIPRANTIS
Purdue University, West Lafayette, U.S.A.
and
OWEN BURKINSHAW
IUPUI, Indianapolis, U.S.A.
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-5007-0 (HB)
ISBN-13 978-1-4020-5007-7 (HB)
ISBN-10 1-4020-5008-9 ( e-book)
ISBN-13 978-1-4020-5008-4 (e-book)
Published by Springer,
P.O. Box 17, 3300 AADordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
All Rights Reserved
© 2006 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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Printed in the Netherlands.
Dedication
To my high school mathematics teachers,
(cid:1)
Aγγ(cid:2)λoν Mατζαβ´ινoν και ∆ηµη´τριoν Kατσω´νην
Charalambos D. Aliprantis
To: Alex, Emily, Joshua, and Andrew
Grandchildren bring new joys to your life.
Owen Burkinshaw
Contents
Foreword ix
Historical Foreword xi
Preface xiii
Acknowledgments xv
List of Special Symbols xvii
Chapter 1. The Order Structure of Positive Operators 1
§1.1. Basic Properties of Positive Operators 1
§1.2. Extensions of Positive Operators 23
§1.3. Order Projections 31
§1.4. Order Continuous Operators 45
§1.5. Positive Linear Functionals 58
Chapter 2. Components, Homomorphisms, and Orthomorphisms 79
§2.1. The Components of a Positive Operator 80
§2.2. Lattice Homomorphisms 93
§2.3. Orthomorphisms 111
Chapter 3. Topological Considerations 133
§3.1. Topological Vector Spaces 133
§3.2. Weak Topologies on Banach Spaces 156
§3.3. Locally Convex-Solid Riesz Spaces 169
vii
viii Contents
Chapter 4. Banach Lattices 181
§4.1. Banach Lattices with Order Continuous Norms 181
§4.2. Weak Compactness in Banach Lattices 207
§4.3. Embedding Banach Spaces 221
§4.4. Banach Lattices of Operators 254
Chapter 5. Compactness Properties of Positive Operators 273
§5.1. Compact Operators 273
§5.2. Weakly Compact Operators 290
§5.3. L- and M-weakly Compact Operators 316
§5.4. Dunford–Pettis Operators 340
Bibliography 359
Monographs 369
Index 371
Foreword
This monograph is a reprint of our original book Positive Operators pub-
lished in 1985 as volume #119 in the Pure and Applied Mathematics series
of Academic Press. With the exception of correcting several misprints and a
few mathematical errors, this edition of the book is identical to the original
one.
At the end of the book we have listed a collection of monographs that
were published after its 1985 publication that contain material and new
developments related to the subject matter of this book.
West Lafayette and Indianapolis CHARALAMBOS D. ALIPRANTIS
May, 2006 OWEN BURKINSHAW
ix
Historical Foreword
(cid:1)
Oλβιos o´στιs ιστoρ´ιηs (cid:2)´σχ(cid:2) µα´θησιν.
Eυριπ´ιδηs
Wise is the person who knows history.
Euripides
Positiveoperatorsmadetheirdebutatthebeginningofthenineteenthcentury.
They were tied to integral operators (whose study triggered the birth of functional
analysis) and to matrices with nonnegative entries. However, positive operators
were investigated in a systematic manner much later. Their study followed closely
the development of Riesz spaces. An address of F. Riesz in 1928 On the decom-
position of linear functionals [166] (into their positive and negative parts), at the
InternationalCongressofMathematiciansinBologna,Italy,markedthebeginnings
ofthestudyofRieszspacesandpositiveoperators. ThetheoryofRieszspaceswas
developed axiomatically in the mid-1930s by H. Freudenthal [67] and L. V. Kan-
torovich[85, 86, 87, 88, 89, 90, 91]. Positiveoperatorswerealsointroducedand
studied in the mid-1930s by L. V. Kantorovich, and they made their first textbook
appearance in the 1940 edition of G. Birkhoff’s book Lattice Theory [37]. Un-
doubtedly, the systematic study of positive operators was originated in the 1930s
by F. Riesz, L. V. Kantorovich and G. Birkhoff.
In the 1940s and early 1950s one finds very few papers on positive operators.
In this period, the main contributions came from the Soviet school (L. V. Kan-
torovich,M.G.Krein,A.G.Pinsker,M.A.Rutman,B.Z.Vulikh)andtheJapanese
school (H. Nakano, K. Yosida, T. Ogasawara, and their students). In the 1950 the
book Functional Analysis in Partially Ordered Spaces [92] by L. V. Kantorovich,
B. Z. Vulikh, and A. G. Pinsker appeared in the Soviet literature. This book (that
xi
xii Historical Foreword
has not been translated into English) contained an excellent treatment, up to that
date, of positive operators and their applications.
Sincethemid-1950s,researchonpositiveoperatorshasgainedconsiderablemo-
mentum. From1955to1970importantcontributionscamefromT.Andoˆ[20]–[24],
C. Goffman [70], S. Kaplan [93], S. Karlin [95], P. P. Korovkin [98], M. A. Kras-
noselskii, P. P. Zabreiko, E. I. Pustylnik, and P. E. Sobolevskii [100], U. Kren-
gel[105, 106],G.Ya.Lozanovsky[119]–[123],W.A.J.LuxemburgandA.C.Za-
anen [125, 130], H. Nakano [148]–[152], I. Namioka [153], A. Peressini [162],
H. H. Schaefer [171], and B. Z. Vulikh [189]. Thus, by the end of the 1960s the
“ground work” for the theory of positive operators was well established.
The 1970s can be characterized as the “maturity period” for the theory of
positive operators. In 1974, the first monograph devoted entirely to the subject
appeared in the literature. This was H. H. Schaefer’s book Banach Lattices and
Positive Operators [174], whoseinfluenceonthedevelopmentofpositiveoperators
was enormous. By this time, the number of mathematicians working in the field
had increased, and research was carried out in a more systematic manner. The
growth of the subject has been very fast; and, in addition, applications to other
disciplineshavestartedtoflourish. Anothermilestoneforthe1970swasthebreak-
throughpaperofP.G.DoddsandD.H.Fremlin[54]onpositivecompactoperators.
The list of contributors to the theory of positive operators in the 1970s includes
Y. A. Abramovich, C. D. Aliprantis, S. J. Bernau, A. V. Buhvalov, O. Burkin-
shaw, D. I. Cartwright, P. G. Dodds, M. Duhoux, P. van Eldik, J. J. Grobler,
D. H. Fremlin, H. P. Lotz, W. A. J. Luxemburg, M. Meyer, P. Meyer-Nieberg,
R.J.Nagel,U.Schlotterbeck,H.H.Schaefer,A.R.Schep,C.T.Tucker,A.I.Vek-
sler, A. W. Wickstead, M. Wolff, and A. C. Zaanen.
The 1980s have started very well for the theory of positive operators. A series
of important papers on the subject have been written by the authors [9, 10, 11,
12, 13, 14, 15, 16, 17]. Another excellent book on positive operators was added
to the literature of positive operators. This was A. C. Zaanen’s book Riesz Spaces
II [197] dealing primarily with research done up to 1980. In addition, a pool
of talented young mathematicians had joined the ranks of researchers of positive
operators; many have already made important contributions. Also, moreand more
mathematiciansofthetheoryofBanachspacesarestudyingpositiveoperators,and
this was given an extra boost to the subject. In addition, the theory of positive
operatorshasfoundsomeimpressiveapplicationsinavarietyofdisciplines,ranging
from mathematical physics to economics.
Thus,asweareprogressingintothe1980sthefutureofpositiveoperatorslooks
bright. The field is alive and grows both in theory and applications. It is our hope
that you, the reader, will also contribute towards this growth.
Indianapolis CHARALAMBOS D. ALIPRANTIS
May, 1984 OWEN BURKINSHAW
Preface
Linear operators have been studied in various contexts and settings in the
past. Their study is a subject of great importance both to mathematics
and to its applications. The present book deals mainly with the special
class of linear operators known as positive operators. A linear operator
between two ordered vector spaces that carries positive elements to posi-
tive elements is referred to as a positive operator. For instance, the linear
operator T: C[0,1] → C[0,1], defined by
(cid:1)
x
Tf(x) = f(t)dt,
0
carries positive functions of C[0,1] to positive functions, and is thus an ex-
ample of a positive operator. The material covered in the book has some
overlapwiththematerialinthebooksbyH.H.Schaefer[174]andA.C.Zaa-
nen [197]. However, our attention is mostly focused on recent developments
of the subject and the overlap with the above-mentioned books is kept to
a minimum. On the other hand, we have intentionally covered those topics
where the ingredient of positivity allowed us to obtain beautiful and elegant
results.
It is well known that many linear operators between Banach spaces aris-
inginclassicalanalysisareinfactpositiveoperators. Forthisreason, inthis
bookpositiveoperatorsarestudiedinthesettingofRieszspacesandBanach
lattices. In order to make the book as self-sufficient as possible, some basic
resultsfromthetheoryofRieszspacesandBanachlatticesareincludedwith
proofs as needed. On the other hand, we assume that the reader is familiar
with the elementary concepts of real analysis and functional analysis.
xiii