Table Of ContentSpringer Optimization and Its Applications 146
Themistocles M. Rassias
Valentin A. Zagrebnov Editors
Analysis
and Operator
Theory
Dedicated in Memory of Tosio Kato’s
100th Birthday
Foreword by Barry Simon
Springer Optimization and Its Applications
Volume 146
Managing Editor
Panos M. Pardalos (University of Florida)
Honorary Editor
Ding-Zhu Du (University of Texas at Dallas)
Advisory Editors
J. Birge, Booth School of Business (University of Chicago)
S. Butenko, Department of Industrial and Systems Engineering (Texas A & M
University)
F. Giannessi, Dipto. Matematica (University of Pisa)
S.Rebennack,InstituteofOperationsResearch(KarlsruheInstituteofTechnology)
T.Terlaky,DepartmentofIndustrialandSystemsEngineering(LehighUniversity)
Y. Ye, Department of Engineering (Stanford University)
Aims and Scope
Optimization has beenexpanding inall directions atanastonishing rate duringthe
lastfewdecades.Newalgorithmicandtheoreticaltechniqueshavebeendeveloped,
thediffusionintootherdisciplineshasproceededatarapidpace,andourknowledge
ofallaspectsofthefieldhasgrownevenmoreprofound.Atthesametime,oneof
themoststrikingtrendsinoptimizationistheconstantlyincreasingemphasisonthe
interdisciplinary nature of the field. Optimization has been a basic tool in all areas
of applied mathematics, engineering,medicine, economics andother sciences.
The series Springer Optimization and Its Applications publishes undergraduate
andgraduatetextbooks,monographsandstate-of-the-artexpositoryworksthatfocus
on algorithms for solving optimization problems and also study applications
involvingsuchproblems.Someofthetopicscoveredincludenonlinearoptimization
(convex and nonconvex), network flow problems, stochastic optimization, optimal
control, discrete optimization, multi-objective programming, description of soft-
ware packages, approximation techniques and heuristic approaches.
More information about this series at http://www.springer.com/series/7393
Themistocles M. Rassias
(cid:129)
Valentin A. Zagrebnov
Editors
Analysis and Operator
Theory
’
Dedicated in Memory of Tosio Kato s 100th
Birthday
Foreword by Barry Simon, IBM Professor of Mathematics
and Theoretical Physics, Emeritus; California Institute
of Technology, Pasadena, CA 91125, USA
123
Editors
Themistocles M.Rassias Valentin A.Zagrebnov
National Technical University of Athens Institute of Mathematics
Zografou Campus Aix-Marseille University
Athens, Greece Marseille, France
ISSN 1931-6828 ISSN 1931-6836 (electronic)
SpringerOptimization andIts Applications
ISBN978-3-030-12660-5 ISBN978-3-030-12661-2 (eBook)
https://doi.org/10.1007/978-3-030-12661-2
LibraryofCongressControlNumber:2019930373
MathematicsSubjectClassification(2010): 46L80,47A58,46Txx,00A79
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Foreword
TosioKato(1917–1999)profoundlyinfluencedthescienceofhistime.Hewasthe
founder of a large field, the theory of Schrödinger operators (which describes the
mathematics of the fundamental objects of the physics of ordinary matter at small
scale). Indeed, one could say collection of fields—for example, shortly after of
writing this, I will be attending a conference on almost periodic and random
Schrödinger operators.
Kato not only founded the field with his 1951 paper on self-adjointness. He
continuedwithinfluentialcontributionsfortherestofhislife,especiallythenext30
years, on a variety of aspects of the theory including perturbation and scattering
theory, spectral analysis, properties of eigenfunctions, and adiabatic theory. His
work is marked by depth, beauty, and elegance.
Like Newton who also made pivotal contribution to Coulomb systems, Kato
found his great self-adjointness result while evacuated from his University and its
library. Because of the War, he spent the middle part of the 1940s in the Japanese
countryside where he also found his basic results on perturbation theory. This
disruption explains why Kato only received his doctorate when he was over 30
years of age.
Like many of the other founders of modern mathematical physics (e.g., Jost,
Thirring, Wightman), Kato’s formal training was in physics at the University of
Tokyo,whereheservedasaProfessorofPhysics,buthelearnedtoexploitrigorous
proof and spent the latter half of his career as a Professor of Mathematics at
University of California, Berkeley. Kato’s opus wasn’t limited to Schrödinger
operators. He also made seminal contributions to the theory of nonlinear PDEs
where he was an early pioneer. Besides these two topics, he has worked in semi-
group theory and to a variety of parts offunctional analysis.
Thisbook,writtenbyagroupofadmirersofKatowhowereallinfluencedbyhis
work, was begun during the centennial year of Kato’s birth and is dedicated to his
memory. Its breadth mirrors his.
Speakingformyself,IhavebeenimpactedbyKato’sworkthroughoutmycareer
up until today. As a graduate student 50 years ago, I learned an enormous amount
from his great book on Perturbation Theory which was central to my first major
v
vi Foreword
work on the quantum anharmonic oscillator. During the 1970s, we exchanged
numerous letters which stimulated research on both our parts.
I am very glad to be able to honor Kato with my contribution here and would
like to thank the editors for putting this project together, for inviting me to par-
ticipate and asking me to write this foreword.
Los Angeles, USA Barry Simon
October 2018 IBM Professor of Mathematics and
Theoretical Physics, Emeritus;
California Institute of Technology
Pasadena, CA 91125, USA
e-mail: [email protected]
Preface
Analysis and Operator Theory—Dedicated in Memory of Tosio Kato’s 100th
Birthday features a collection of carefully selected research as well as survey
articles devoted to a broad spectrum of subjects of Mathematical Analysis and
Mathematical Physics, to which Kato contributed monumental results.
The breadth and full depth of Kato’s legacy is impossible to exhaust by even
more than one volume dedicated to him. A brilliant mathematician, world-
renowned specialist in functional analysis and operator theory, he was also an
outstandingteacher.Hisgreatbook“PerturbationTheoryforLinearOperators”first
published by Springer in 1966 is an encyclopedia for mathematicians and mathe-
matical physicists. Many of his papers, e.g., on operator theory, evolution equa-
tions,orspectralanalysis,aresoremarkablywrittenthattheyarestillgreatclassics
to be read in the original.
The book chapters presented here have been written by a number of experts in
the various areas of Kato’s interest. They are aiming to present the actual state
oftheartinthetopics,whichweredevelopedbyTosioKatoinvariousbranchesof
analysisandoperatortheory.Someofhisachievements,suchasKato’sinequality,
the Kato type matrix limit theorem, the Howland–Kato commutator problem, the
Kato class of potentials, or the Trotter–Kato product formulae, are directly mani-
festinginthetitlesofthechaptersofthisvolume.Amongothers,therearerelatedto
or inspired by certain ideas of Kato.
A special article presents a report of Tosio Kato’s work on nonrelativistic
QuantumMechanics,andoneof thechaptersisdedicatedtoanunpublished paper
of his.
Itishopedthatthispublicationprovidesanextensiveaccountofresearchresults
which will be of usefulness for a wide readership, from graduate students to
established researchers in the corresponding domains.
vii
viii Preface
We would like to express our gratitude to all the scientists who contributed
valuable works in this book as well as to Barry Simon who wrote the foreword.
Additionally, we would like to acknowledge the superb assistance of the staff of
Springer for the preparation of this publication.
Athens, Greece Themistocles M. Rassias
Marseilles, France Valentin A. Zagrebnov
Contents
Complementarity and Stochastic Independence . . . . . . . . . . . . . . . . . . . 1
Luigi Accardi and Yun-Gang Lu
Norm Conditions for Separability in MMm(cid:1)(cid:1) MMn . . . . . . . . . . . . . . . . . . . 35
Tsuyoshi Ando
Kato’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
W. Arendt and A. F. M. ter Elst
Tosio Kato’s Unpublished Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Claude Bardos and Hisashi Okamoto
On the Border Lines Between the Regions of Distinct Solution Type
for Solutions of the Friedmann Equation. . . . . . . . . . . . . . . . . . . . . . . . 65
Hellmut Baumgärtel
Scattering on Leaky Wires in Dimension Three. . . . . . . . . . . . . . . . . . . 81
Pavel Exner and Sylwia Kondej
Computing Traces, Determinants, and ff-Functions
for Sturm–Liouville Operators: A Survey . . . . . . . . . . . . . . . . . . . . . . . 93
Fritz Gesztesy and Klaus Kirsten
On the Domain of a Magnetic Schrödinger Operator
with Complex Electric Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Bernard Helffer and Jean Nourrigat
Matrix Limit Theorems of Kato Type Related to Positive
Linear Maps and Operator Means. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Fumio Hiai
The Howland–Kato Commutator Problem. . . . . . . . . . . . . . . . . . . . . . . 191
Ira Herbst and Thomas L. Kriete
ix
