Table Of ContentRichardM.Aron,EvaA.GallardoGutiérrez,MiguelMartin,DmitryRyabogin,
IlyaM.Spitkovsky,ArtemZvavitch(Eds.)
TheMathematicalLegacyofVictorLomonosov
Advances in Analysis and
Geometry
|
Editor-in-Chief
Jie Xiao, Memorial University, Canada
EditorialBoard
Der-ChenChang,GeorgetownUniversity,USA
GoongChen,TexasA&MUniversity,USA
AndreaColesanti,UniversityofFlorence,Italy
RobertMcCann,UniversityofToronto,Canada
De-QiZhang,NationalUniversityofSingapore,Singapore
KeheZhu,UniversityatAlbany,USA
Volume 2
The Mathematical
Legacy of Victor
Lomonosov
|
Operator Theory
Edited by
Richard M. Aron, Eva A. Gallardo Gutiérrez,
Miguel Martin, Dmitry Ryabogin, Ilya M.Spitkovsky, and
Artem Zvavitch
Editors
Prof.RichardM.Aron Prof.DmitryRyabogin
KentStateUniversity KentStateUniversity
DepartmentofMathematicalSciences DepartmentofMathematicalSciences
800E.SummitSt.Kent 800E.SummitSt.Kent
Kent,Ohio44242 Kent,Ohio44242
USA USA
[email protected] [email protected]
Prof.EvaA.GallardoGutiérrez Prof.IlyaM.Spitkovsky
UniversidadComplutensedeMadrid NewYorkUniversityAbuDhabi
FacultaddeMatemáticas SaadiyatIsland
PlazadeCiencias3 DivisionofScience
28040Madrid 129188AbuDhabi
Spain UAE
[email protected] [email protected]
Prof.MiguelMartin Prof.ArtemZvavitch
UniversidaddeGranada KentStateUniversity
Dept.deAnalisisMatematico DepartmentofMathematicalSciences
CampusFuentenuevaS/N 800E.SummitSt.Kent
18071Granada Kent,Ohio44242
Spain USA
[email protected] [email protected]
ISBN978-3-11-065339-7
e-ISBN(PDF)978-3-11-065675-6
e-ISBN(EPUB)978-3-11-065346-5
ISSN2511-0438
LibraryofCongressControlNumber:2020935870
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2020WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Editors’ Introduction
We introduce this volume with a mixture of feelings: Fond memories for Victor
Lomonosov as a colleague and friend with a marvelous sense of humor, but also
afeelingofgreatsadnessforthelossofapersonpossessingsuchmathematicalbril-
liance. Reading—or rereading—his work, one is struck by its depth, elegance, and
apparentsimplicity.Itiseasytobefooledbythebrevityofhiswork:Forexample,
therearebarely15pagesintotalinhis1973paperontheinvariantsubspaceproblem,
the1991IsraelJournalpaperontheBurnsidetheoremininfinitedimensions,andthe
2000IsraelJournalpaperontheBishop–PhelpstheoremincomplexBanachspaces.
However,theseclearlywrittenpapers,inabsolutelycentralareasofoperatortheory
andfunctionalanalysis,areindeedveryprofoundandhavehadasignificantimpact
onthedevelopmentofanalysisduringthelasthalfcentury.
Wethankthemanymathematicianswhohavecontributedtheirexcellentworkto
celebrateandmemorializeVictorLomonosov’scontributionstooperatortheoryand
functionalanalysis.WearealsogratefultoDeGruyterforitssupportofthisproject.
February2020 RichardM.Aron
EvaA.GallardoGutiérrez
MiguelMartín
DmitryRyabogin
IlyaM.Spitkovsky
ArtemZvavitch
(editors)
https://doi.org/10.1515/9783110656756-201
Per Enflo’s personal thoughts about
Victor Lomonosov
Earlyin1973,therewasexcitementinthemathematicalworld.Ayoungmathemati-
cian from Russia, Victor Lomonosov, had proved that operators commuting with a
compact operator on a Banach space have invariant subspaces. The short, elegant
proofhadmade40yearsofdevelopmentobsolete.IwasatUCBerkeleyatthattime.
I remember that the leading operator theorists there immediately started to work
hardtoseeifthenewpowerfultechniquecouldsolvetheentireinvariantsubspace
problem. It seemed hopeful that this was within reach, since Lomonosov’s proof
evenshowed,thatanoperatorcommutingwithanoperatorwhichcommuteswitha
compactoperatorhasinvariantsubspaces.Theydidnotsucceed,butevensoitwas
clearthatLomonosov’snewideashadalreadyforeverchangedthefieldofoperator
theory.
WhowasVictorLomonosov?ItseemedthatnobodyatBerkeleyhadmethimor
knewanybodywhohadmethim.Coulditevenbethat“Lomonosov”wasamadeup
nameforthecollaborationofseveralmathematicians,liketheFrench“Bourbaki”?
Fifteenyearslater,intheearlysummerof1988,whenIwassittinginmyoffice
atKentStateUniversity,mytelephonerang.ThepersonintroducedhimselfasVictor
Lomonosov.HewasvisitingafriendintheUS.Thefriendquicklytookoverthetele-
phoneconversationandIlearnedthatVictorLomonosovhadplanstoreturnandem-
igratetotheUSinearly1989.Tome,thisphonecallwasextremelysurprising,almost
spooky,andatthesametimeveryexciting.SoIaskedVictortocontactmeimmedi-
atelywhenhewasbackintheUS.
WhenVictorcamebackinearly1989,Itriedallregularwaystofindsomeshort
termemploymentforhimatKentStateUniversity.Buttherewasnomoneyanywhere.
IknewthatPresidentMichaelSchwartzhadtheambitiontomakeKentStateatop
levelresearchuniversityandIknewthathehadfaithinme.SoIwenttoareception
whichhewasgoingtoattend.ItookhimasideandItoldhimthatKentStatehadan
opportunitytohireamathematicianwhohadjustcomeoutoftheSovietUnion,and
whohaddonesomesensational,groundbreakingwork.Althoughthisworkwas15
yearsinthepast,chanceswerethathewasstillatoplevelmathematician.Michael
Schwartzremarkedthattheremightbepoliticalreasonsforthelongsilenceandthen
heaskedmethreequestions:“DoeshespeakEnglish?”Ianswered:“Heseemsvery
ambitioustolearntospeakbetter.”Nextquestion:“Isheasgoodasyouare?”BeforeI
hadcommented,hecontinuedwiththethirdquestion:“Doyouwanthimhere?”SoI
said:“YES!”Thenhesaid:“Wewillgethim!”Andtwodayslater,therewasmoneyto
hireVictorfor6months.Andlongbeforetheendofthese6months,itwasclearthat
KentStatehadmadeagreatdecision.Victorgotoffersfromseveralotheruniversities,
buthestayedatKentState.Andovertheyearshemadeaveryimportantcontribution
https://doi.org/10.1515/9783110656756-202
VIII | PerEnflo’spersonalthoughtsaboutVictorLomonosov
tothemathematicallifeofKentStateUniversity,mainlythroughhisownworkbut
alsothroughhismanycontactswithmathematiciansfromtheformerSovietUnion.
Not surprisingly, the cultural differences between the Soviet Union and the US
sometimesshowedupinthebeginningofVictor’sstay.Hereisoneexample:
AnAmericantoVictor:“Oh,youdonothavesocialsecuritynumbersintheSoviet
Union.Buthowdoyouthencountpeople?”
Victor:“One,two,three...”
ImetalmostdailywithVictorinthebeginningofhisstay.Weusuallyhadlunchat
BurgerKing,andwediscusseddailylifeissuesandmathematics.Victorworkedhard
toadjusttoAmericanlife.Gettingadrivinglicenseasquicklyaspossiblewasimpor-
tant.HebroughthisEnglish-Russiandictionarytothewrittentestandwasallowedto
useit“aslongasitdoesnothaveanyOhiotrafficlawsinit.”
And he worked hard to improve his English, by talking, reading and watching
TV.Intheyearsaround1990,perhapsinconnectionwiththefinancialcrisis,ithad
becomecommontoblamestudents’failuresontheirprofessors’poorEnglish.So,for
VictorgettingatenuredpositionatKentState,itwasclearthatthiscouldbeapotential
issue.But,withVictor’sfastlearning,nothinghappenedlikethat.
Atthesametime,asVictoradjustedtoalifeintheUS,hedevelopedhisdeep,
ingenious,andbeautifultheoryofoperatoralgebrasonBanachspaces.Itisbasedon
aninequalitywhich,whenspecializedtofinitedimensions,givesBurnside’sclassical
theorem.And,besidesstrengtheningandimprovingmanyearlierresultsonoperator
algebrasandinvariantsubspaces,itsuggeststhefollowing,insomesenseultimate,
conjecturefora“positive”answertotheinvariantsubspaceproblem:IfAisanopera-
∗
toronaBanachspace,thenA hasaninvariantsubspace.Inlaterwork,hedeveloped
furtherhisworkonextensionsofBurnside’stheorem.
Victor’s remarkable ability to find new approaches to difficult problems led to
another great triumph, when he solved the long-standing problem of whether the
Bishop–Phelps’theoremholdsforBanachspacesoverthecomplexfield.Oneversion
ofthecelebratedBishop–Phelps’theoremfromtheearly1960sstatesthat,inaBa-
nachspaceovertherealnumbers,aclosed,bounded,convexsethasasupportpoint,
thatis,apointwheresomefunctionalattainsitssupremum.
In1977,J.BourgainprovedthatforBanachspaceswiththeRadon–Nikodymprop-
erty,thisholdsalsoforBanachspacesoverthecomplexfield.
In2000,Victorgaveanexampleofaclosed,bounded,convexsetwithoutsupport
pointsinaBanachspaceoverthecomplexfield.Andinasubsequentpaperheshowed
thatthecomplexversionofBishop–Phelps’theoremfailsineveryspacewhichisa
predualofanyuniformnonself-adjointdualoperatoralgebra.So,hiscounterexample
isnotjustanisolated“pathology,”butratheritispartofamoregeneralphenomenon.
AsIalreadymentioned,Victorhadaremarkableabilitytofindnewapproachesto
difficultproblems.Inhisownwork,hewaslookingfornew,powerfulideas.Andhis
ownpapersareusuallynotverylong.Thepaper“Exponentialnumbersoflinearop-
eratorsinnormedspaces”startedasa4-pagepaperbyVictorandmyself.Wethought
PerEnflo’spersonalthoughtsaboutVictorLomonosov | IX
thatthepaperhadaniceidea,worthpublishing.V.Gurariyjoinedus,andthepaper
grewtoa15-pagepaper.ThenYu.I.Lyubichjoinedusanditwaseventuallypublished
asa35-pagepaper.
Victor frequently collaborated with other mathematicians. In his joint publica-
tions—papersofhighquality—thereisabroadspanoftopics,fromanalysistoalmost
purealgebra.Mydiscussionswithhimfellmostlywithintheareaofoperatortheory.
Andforme,theywereanimportantinspiration,bothformyowneffortsandformy
workwithdoctoralstudents.Ourdiscussionscontinuedtotheendofhislife.Iremem-
berthat,shortlybeforehispassing,weconsideredthefollowingquestiontowhich
noneofushadananswer:Consideraone-to-oneoperatorT withdenserange.For
∗
whichTisthereanoperatorV,similartoT,suchthatVandV havethesamerange?
SuchthatV andV∗havedisjointranges,exceptfor{0}?
Victor’sworkhashadagreatimpact.Severalofhisresultsarenowclassicaland
parts of standard courses in functional analysis. They are famous, not just among
functionalanalysts,butintheentiremathematicalworld.
Östervåla PerEnflo
5/282019
PhotoofVictorLomonosov
