Table Of ContentFakultät Mathematik und Naturwissenschaften Fachrichtung Mathematik Institut für Analysis
Proceedings of the Conference
Positivity IV − Theory and Applications
Dresden, July 25 − 29, 2005
Impressum
Publisher:
Technische Universität Dresden
Institut für Analysis
Editors:
Prof. Dr. Martin R. Weber
Prof. Dr. Jürgen Voigt
Layout:
Ingo Tschichholtz, Helga Mettke
Cover:
Helga Mettke
Printing:
Copy Cabana Steffen Kürbis & Paul-Stefan Scholz GbR, Dresden
Contact:
Prof. Dr. Martin R. Weber
Institute of Analysis
Department of Mathematics
Faculty of Science
Technische Universität Dresden
01062 Dresden (Germany
Tel.: ++49 (0351) 4 63 35434 / Fax: ++49 (0351) 4 63 37202
e-mail: [email protected]
All articles are refereed.
Cover Picture: Helga Mettke
No part of this publication may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, without the prior permission of the copyright owner.
ISBN 3-86005-512-7
May 2006
Proceedings of the Conference
Positivity IV − Theory and Applications
Dresden (Germany), July 25 - 29, 2005
Organizing Committee: Martin R. Weber, Jürgen Voigt
Anke Kalauch, Ingo Tzschichholtz
Helga Mettke
Support from the following institutions is gratefully acknowledged:
• Deutsche Forschungsgemeinschaft (DFG)
• Sächsisches Staatsministerium für Wissenschaft und Kunst
• Gesellschaft von Freunden und Förderern der TU Dresden e.V.
• Technische Universität Dresden.
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Foreword
TheInternational Conference
Positivity IV – TheoryandApplications
was organized at the Department of Mathematics of the Technische Universita¨t Dresden
and took place from July 25 until 29, 2005 in Dresden (Germany). It was already the fourth
conferencedevotedtothetopicofPositivityaftertheconferencesheldinAnkara(Turkey)1998,
Nijmegen (Netherlands)2001 andRhodos (Greece)2003.
In many fields of mathematical thinking, but also in the wider field of science, economics
and engineering and their interaction the property of positive quantities, functions, solutions
and others is well defined and of particular importance. The most abstract formulation of posi-
tivity makes use of the positive reals and was systematically developed in the last eighty years,
although the idea of positivity is much older. It has turned out that for many purposes vec-
tor spaces equipped with a partial order (or equivalently, with a convex cone), ordered normed
spaces and the corresponding classes of positive operators form the appropriate setting for re-
search and application. The latter establishes nowadays a considerable independent part of
functional analysis very interacting with many mathematical disciplines such as optimization,
mathematicaleconomy,semigrouptheory,measureandintegrationtheory,stochastics,ordinary
andpartialdifferential equationsand numericalmethods.
Altogether 44 mathematicians from Europe, Africa, America and Asia participated in the
conference. Thecompletelistisincludedasoneofthenextpages. In11plenarylecturesand28
short communications a large spectrum of the ideas of positivity asa main stream in functional
analysiswascovered.
TheProceedingsoftheconferencepresentanoverviewofthetopicsdiscussedandcontains
paperssubmittedbytheparticipants. TheeditorsoftheProceedingsthankallpersonswhowere
engaged to host the conference and to organize the social events, and also thank all colleagues
whowereinvolvedintherefereeingprocedure,inparticular,IngoTzschichholtz,AnkeKalauch
andHelga Mettke.
Many of the participants visited the city of Dresden for the first time. So a guided evening
walk through the center of Dresden was of great interest the more that the rebuilding of the
famous Frauenkirche Church was nearly finished and so the historical center of Dresden could
beadmired in itsfullbeauty.
In the middle of the conference week the participants and accompanying persons went by
train to Rathen, a small village in the Saxon Switzerland, about 40 km south of Dresden. From
thereasmallsteepstraining path leadthrough thetypicalpillar-like mountains upto theBastei
Plateau, where all of us were compensated for our exertions by the panorama of the so-called
Elbe-sandstonemountains.
IV Foreword
The conference was supported by Deutsche Forschungsgemeinschaft (DFG), the regional
Saxon State Ministry for Science and Art, the Department of Mathematics and the Society of
FriendsandSupporters oftheTechnischeUniversita¨tDresden.
We appreciated continuing the tradition of conferences on positivity and hope to meet next
time inBelfast.
Dresden,May2006 MartinR.Weberand Ju¨rgenVoigt
Contents
III Foreword
1 Opening speech of the Vice-Dean for Mathematics, Prof. Dr. Volker Nollau
Contributions
5 ALEKHNO, E. A.
Spectral Properties of Band Irreducible Operators
15 ALPAY, Şafak and MISIRLIOĞLU, Tunç
Invariant Subspace Theorems for Families on a Banach Lattice
21 AMOR, Fethi Ben
The Riesz-Kantorovich formula for the modulus of a complex order bounded
disjointness preserving operator
29 BOULABIAR, Karim and BUSKES, Gerard
A note on bijective disjointness preserving operators
35 BUSKES, Gerard and REDFIELD, R. H.
Band preserving operators on lattice-ordered groups
45 CHEN, Z. L.
On the order continuity of the regular norm
53 EMEL’YANOV, E. Yu.
Positive Asymptotically Regular Operators in L1-spaces and KB-spaces
63 GROBLER, J. J.
Bivariate and marginal function spaces
73 KITOVER, A. K. and WICKSTEAD, A. W.
Positive operators without invariant sublattices
79 KÖNIG, Heinz
Stochastic Processes on the Basis of New Measure Theory
93 KUSRAEV, A. G. and KUTATELADZE, S. S.
Boolean Valued Analysis and Positivity
107 METAFUNE, Giorgio and RHANDI, Abdelaziz
The dominant eigenvalue of nonsymmetric elliptic operators with Dirichlet
boundary conditions
115 dePAGTER, B. and RICKER, W. J.
R-boundedness of C(K)-representations, group homomorphisms, and Banach
space geometry
131 STEIN, Martin and VOIGT, Jürgen
On modulus semigroups and their generators
135 THIEME, H. R. and VOIGT, J.
Stochastic semigroups: their construction by perturbation and approximation
147 VÄTH, Martin
Generalized Ideal Spaces and Applications to the Superposition Operator
155 WEBER, Martin R.
Finite Elements in Vector Lattices
173 WNUK, Witold and WIATROWSKI, Błazej
When are ultrapowers of normed lattices discrete or continuous?
Attachment
183 List of Participants
Opening speech of the Vice-Dean for Mathematics,
Prof. Dr. Volker Nollau
LadiesandGentlemen,
DearColleaguesandGuests
On behalf of the Technical University Dresden, the Faculty of Mathematics and Natural
SciencesandtheDepartment ofMathematicsIwelcomeyou tothe
InternationalConferencePositivity IV.
We are very pleased that the Department of Mathematics of the Technical University Dres-
denwasselectedto hostthis conference.
Let me shortly outline the history and the development of our university and its department
ofmathematics. In 2003 ouruniversity celebratedits175th anniversary.
Our university was founded in 1828 as a Higher Technical Educational Institute. Its develop-
ment was and is closely related to the growth of the regional Saxon and German industry and
economy - in the early years especially to the formation of the Saxon mechanical engineering
industry.
In 1851 the Institute was upgraded to the Royal Saxon Technical School. Later the name
was changed to Technische Hochschule Dresden and in 1961 reached its final designation as
theTechnischeUniversita¨tDresden.
By1993theTechnicalUniversityhadincorporatedvariousotherinstitutionsofteachingand
research, especially the Academy of Forestry, the College of Traffic Science and the Medical
AcademyaswellastheTeacher’sCollege.
AtthepresenttimetheTechnicalUniversityDresdenalsooffersprogramsinhumanitiesand
social sciences as well as in economics, linguistics and other subjects. The total of currently
about 33 000 students is roughly evenly split between these new subjects and the traditional
disciplines suchasengineering, science,forestryandcomputer science.
Since the middle of the nineteenth century Saxony was leading in integrating the teaching
of science into the curriculum of the secondary and higher educational system. Also there has
always been a close interaction between mathematical research and its engineering applica-
tions. In other words, mathematicians working at the university were continually involved in
engineering development and progress. This engagement also shaped their teaching activities.
2 Opening speech of the Vice-Dean for Mathematics, Prof. Dr. Volker Nollau
Consequently,thecurriculumofMathematicsattheTechnischeUniversita¨tDresdenhasalways
requiredaminorin anengineeringorsciencesubject.
Since more than ten years the Department offers the following study tracks Mathematics,
Technomathematics, Economathematicsand TeachingofMathematics.
Moreover, the Department of Mathematics is responsible for the mathematical training in
allotherfacultiesoftheuniversity.
Our graduates have never encountered difficulties in finding suitable employment through-
out Germany and Europe. It is hoped that this pool of highly qualified professionals will also
supportthedevelopment ofthe regionalindustrialbase.
NeverthelesstheDepartmentofMathematicsalsocoverssomemoretheoreticalbranchesof
Mathematics. From the mathematicians who worked at our Department and are well known in
Analysis,Topology andFunctionalAnalysisletmemention only
• O. SCHLO¨MILCH (1823 -1901): among other things known for a general remainder in
Taylor’sfomula
• A. HARNACK (1851-1888): Inequalities for potential functions relevant in partial differ-
entialequations
• H. SEIFERT (1907-1996) and W. THRELFALL (1888-1949): well known for their book
,,LehrbuchderTopologie”,whichwasthefirstmonographdevotedtocombinatorictopol-
ogy
• F. RELLICH (1906-1955): Spectraltheory.
Up to now several mathematicians of the Department in various Institutes have strong rela-
tions tofunctional analysis.
Nowadays the Department is subdivided into the Institutes of Algebra, Analysis, Geom-
etry, Mathematical Stochastics, Numerical Analysis, Scientific Computation and the Chair of
Didactics,eachwith theirownspecialresearchtopics.
WehaveanextendedcooperationwithotheruniversitiesthroughoutEuropeandalloverthe
world. This is also reflected by a remarkably high percentage of partial studies of our Diploma
studentsabroad. Presently,manyDiploma-andPhD-studentsareinvolvedintheresearchwork
ofourinstitutes.
To this meeting the Technische Universita¨t Dresden and the Department of Mathematics
welcome participants from Great Britain, France, the Czech Republic, Poland, the Russian
Federation,Greece,theNetherlands,Slovenia,Turkey,theUnitedStates,Canada,SouthAfrica,
thePeople’sRepublic ofChina, Japan,Morocco,Tunisia andGermany.
The aims of the Conference on Positivity, which I AM PLEASED TO OPEN have been al-
readyspelledoutintheannouncements. Therefore,letmementiononlythatitisveryimportant
