Nigel J. Kalton Selecta: Volume 2

778 Pages·2016·135.5 MB·English

by  Fritz Gesztesy

Contemporary Mathematicians
Fritz Gesztesy
Gilles Godefroy
Loukas Grafakos
Igor Verbitsky
Editors
Nigel J. Kalton
Selecta
Volume 2

Contemporary Mathematicians
Joseph P.S. Kung
Editor
Moreinformationaboutthisseriesathttp://www.springer.com/series/4817
Fritz Gesztesy • Gilles Godefroy (cid:129)
Loukas Grafakos (cid:129) Igor Verbitsky
Editors
Nigel J. Kalton Selecta
Volume 2
Editors
FritzGesztesy GillesGodefroy
DepartmentofMathematics UniversitéPierreetMarieCurie
UniversityofMissouri InstitutdeMathématiquesdeJussieu
Columbia,Missouri,USA Paris,France
LoukasGrafakos IgorVerbitsky
DepartmentofMathematics DepartmentofMathematics
UniversityofMissouri UniversityofMissouri
Columbia,Missouri,USA Columbia,Missouri,USA
ContemporaryMathematicians
ISBN978-3-319-18798-3 ISBN978-3-319-18799-0 (eBook)
DOI10.1007/978-3-319-18799-0
LibraryofCongressControlNumber:2016940248
MathematicsSubjectClassification(2010):00B60,01A75
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Preface
NigelKalton’sresearchspansmorethan40years,andhispublicationsrepre-
sentthousandsofpagesoforiginal,deep,andconciselywrittenmathematics.
Sadly, the Great Book of Analysis composed by Nigel suddenly closed on
August 31, 2010, and we are left with the daunting task of continuing its
legacy. But Nigel’s desire clearly was that, no matter the circumstances,
research must go on. These selected publications and their commentaries
representamodestattempttofulfillthesegoals.
EverysinglearticlethatNigelauthored(orco-authored)containsatleast
one important result, and a fair number of his contributions completely
renewed an entire topic. Therefore, it was quite difficult to select the most
significantarticlesinhis formidablebibliography.Thepresentvolumesaim
toprovidethemathematicalcommunitywithsomeofNigel’smostinfluential
works.Theyincludemostofthemanyfieldstowhichhecontributed.
All mathematicians who were privileged to be Nigel’s collaborators,
colleagues,orstudentsknowwhataprolificproblemsolverhewas.However,
it would not do justice to reduce Nigel’s scientific contributionsto the very
impressive list of open problems that he solved. Indeed, Nigel’s vision of
mathematics was so far-reaching that it enabled him to use quite exotic
objects, such as nonlocally convex spaces, to provide original results in
classical and mainstream analysis. Above all, he was a creative mind who
wouldsetupthemathematicalframeworkinwhichaquestionisbestputinto
itspropercontext,whowouldthenpenetrateitfromallpossibleangles,and
whowouldfinallysolveit,alongwithafamilyofrelatedproblems.Therefore,
his commanding point of view prepared the ground for future work, and
providedhisfollowerswithpreciousnewtools.
This volume is a user’s guide to the tools, techniques, and ideas Nigel
Kaltongaveus.Indeed,leadingexpertskindlyprovidedinsightfulcomments
on each article in our selection. We are mostgratefulto all the contributors
to this volume, as they were instrumental in actually realizing the idea
of Kalton Selecta. Thanks to their work and expertise, this volume is
much more than a mere commemoration of Nigel’s achievements, or a
tribute to his memory. We should rather understand it as the proceedings
of a major scientific event: an attempt to facilitate the understanding of
Nigel Kalton’s contribution,and a starting point for further research on the
v
vi Preface
problems left open. The influence of Nigel’s work is now stronger than
ever, and it is our hope that experienced researchers as well as beginners
will benefitfromeasyaccess to someof hisfundamentalarticlesandideas.
We invite the reader to delve into Nigel Kalton’s work, appreciate every
lineofitandallthatcanbefoundbetweenthelines.Whenenteringthisgold
mine, one mustbe preparedto dig deep, but in return,greatenjoymentand
rewardsshouldbeexpected.
We now proceed to a short description of the contents of the Kalton
Selecta. We decided not to include in these volumes papers co-authored
withourselvesorwithNigel’sdepartmentalcolleagues.Ourintentionwasto
pickarticlesfromthevariousfieldstowhichNigelKaltoncontributed.Each
articleisprecededbyacontributionwrittenbyanexpert,whoisoften,butnot
always,Nigel’scollaborator.Insomecases,thecontributordecidedtogather
commentsonseveralrelatedarticlesinasingletext.Fortheconvenienceof
thereader,thecommentsandthearticleshavebeenclassifiedintoeightparts:
Nonlocally Convex Spaces and Submeasures, DifferentialGames, Operator
Theory,HarmonicAnalysisandPDEs,ApproximationTheory,Geometryand
Banach Spaces, Interpolation Theory, and Probability and Banach Spaces.
Itshould,however,bestressed thatthisclassificationis somewhatarbitrary.
Tightlinksexistbetweenthesevariousfields, andthese linksarefrequently
duetoNigelKaltonhimself.Indeed,Nigelhadanoutstandingabilitytouse
ideasandtechniquesinplaceswheretheywerenotexpectedtobeeffective,
and to relate conceptsthat appeareddisjoint to less penetrating minds. Due
to the great size of the work that we have collected, it was necessary to
split the chapters into two volumes. Nigel’s CV along with the first four
parts comprise the first volume, whereas the second volume consists of the
remainingfourparts.
ThefirstchaptercontainsNigel’sCV,lastupdatedinFebruary2015.PartI
covers a field in which Nigel Kalton is the undisputed leader: Nonlocally
Convex Spaces. This part contains some deep results about the structure
of quasi-Banach spaces, the existence of peculiar objects such as minimal
quasi-Banachspaces,andsubmeasures.ThisvoyagethroughtheKaltonzone
0 (cid:2) p < 1 is commentedon by FernandoAlbiac, José LuisAnsorena, Per
Enflo,DavidFremlin,AnnaKaminska,andBernardMaurey.
PartIIisdevotedtoOptimalControlTheoryanditsrelationshipwithvon
Neumann’sGameTheorythroughDifferentialGames.Thecontributortothis
partisNigel’sco-authorRobertElliott.PartIIIpresentssomecontributionsto
Operatortheory:maximalregularity,Hilbertiantheoryandtraces,similarity
problems,andtherecentandcrucialnotionofR-boundedness.Contributions
weremadebyFritzGesztesy,TamaraKucherenko,GillesLancien,Vladimir
Peller,PierrePortal,andFedorSukochev.
Part IV gathers some articles from Harmonic Analysis and Partial Dif-
ferential Equations and covers various topics, such as lacunarity in har-
monic analysis and H1-functional calculus with its applications to par-
Preface vii
tial differential equations. Comments for this part are written by Sergei
Ivanov, Nikolai Nikolski, Igor Verbitsky, and Lutz Weis. The original arti-
cle “The H1-functional calculus and square function estimates” by N.
Kalton and L. Weis is published as the seventh chapter of this part. This
article has been extensively quoted in the literature and we are grate-
ful to Lutz Weis for his decision to publish this article in the Selecta.
TheothersetofPartsI–IViscontainedinthesecondvolume.PartI(vol-
ume2)concernsApproximationTheory,inthesenseofappliedmathematics,
namelyapproximationbysmoothfunctionsorbyvectorswithfinitesupport
withagivenrateofconvergence.ContributorstothissectionareYuriBrud-
nyi,StephenDilworth,andDenkaKutzarova.PartII (volume2)focuseson
theGeometryofBanachSpaces,atopictowhichNigelcontributedinevery
possibleway:isomorphictheory,isometrictheory,nonlineargeometry... and
ofcourseinterpolationtheory,buttherelevantarticleshaveasectionoftheir
own.ContributorstoPartII(volume2)areYoavBenyamini,JesúsCastillo,
Garth Dales, Anna Kaminska, Gilles Lancien, Mikhail Ostrovskii, Gilles
Pisier,andDirkWerner.
Part III (volume 2) is relevant to interpolation theory and the fasci-
nating links between interpolation lines, twisted sums, quasi-linear maps,
weakerdistancesbetweennon-isomorphicBanachspacesandthenonlocally
convex world. Nigel’s contribution to this field is such that it is fair to
designate the corresponding theory as the Kalton calculus. Contributors to
this section are Michael Cwikel, Mario Milman, Richard Rochberg, and
Stephen Dilworth. Finally, Part IV (volume 2) is devoted to certain aspects
oftheinterplaybetweenProbabilityTheoryandBanachSpaces:decoupling
property,spaces generated by a sequence of independentrandom variables,
andRademacherseries.ContributorstothissectionareStephenDilworthand
StephenMontgomery-Smith.
We conclude this introduction by expressing our deepest gratitude to all
thecontributorstothisvolume.Theircommentsprovidethebesttributethey
could pay to Nigel’s memory:a mathematical analysis of his works. More-
over,wethankthem,andnumerouscolleaguesandfriends,mathematicians,
andpublishers,fortheirencouragement,generousadvice,hands-onhelp,and
steadfastsupportinthepreparationoftheseKaltonSelecta.Inparticular,we
sincerely thank each of the publishers involved for granting us permission
to include their pdf files in these volumes. We are particularly indebted to
Michael Cwikel not only for spearheading the extensive commentaries on
interpolation theory in Part III, but also for supplying us with numerous
comments and correctionsthroughoutthe Selecta. His level of involvement
withthisprojectwentfarbeyondanythingwecouldhavepossiblyexpected.
ManythanksareduetoBirkhäuserSpringerVerlagandtheextraordinary
supportofferedbyThomasHempfling,AnnaMaetzener,KarinNeidhart,and
ChristianeTretter.We alsoexpressourspecialgratitudetoJennyKaltonfor
allherhelpandsupportthroughouttheentireproject.
viii Preface
Finally,wededicatethisbooktoNigelKalton.ThankyouNigel,foralife
devotedtomathematics.
Columbia,Missouri,USA FritzGesztesy
Paris,France GillesGodefroy
Columbia,Missouri,USA LoukasGrafakos
Columbia,Missouri,USA IgorVerbitsky
Contents
PartI ApproximationTheory
Polynomial approximation on convex subsets of Rn
byY.A.BrudnyiandN.J.Kalton.............................. 3
Y.A. Brudnyi: Comments on the Paper “Polynomial
approximationonconvexsubsetsofRn”byY.A.Brudnyiand
N.J.Kalton,Constr.Approx.16(2000),161–199................. 43
Thethresholdinggreedyalgorithm,greedybases,anddualityby
S.J.Dilworth,N.J.Kalton,D.Kutzarova,andV.N.Temlyakov.... 49
S.J. Dilworth: Comments on the Paper “The thresholding
greedyalgorithm,greedybases,andduality”byS.J.Dilworth,
N.J.Kalton,D.Kutzarova,andV.N.Temlyakov,Constr.Approx.
19(2003),575–597........................................... 73
ConvergenceoftheweakdualgreedyalgorithminL -spacesby
p
M.GanichevandN.J.Kalton................................. 79
S.J. Dilworth: Comments on the Paper “Convergence
oftheweakdualgreedyalgorithminL -spaces”byM.Ganichev
p
andN.J.Kalton,J.Approx.Theory124(2003),89–95............ 87
Onthe existenceof almostgreedybases inBanachspacesby
S.J.Dilworth,N.J.KaltonandD.Kutzarova.................... 93
D.Kutzarova:CommentsonthePaper“Ontheexistenceofalmost
greedybasesinBanachspaces”byS.J.Dilworth,N.J.Kaltonand
D.Kutzarova,StudiaMath.159(1)(2003),67–101.............. 129
PartII GeometryofBanachSpaces
TheuniformstructureofBanachspacesbyN.J.Kalton.......... 135
Y.Benyamini:CommentsonthePaper“Theuniformstructure
of Banach spaces” by N.J. Kalton, Math. Ann. 354 (2012),
1247–1288.................................................. 178
ix