Narrow Operators on Function Spaces and Vector Lattices

333 Pages·2012·1.621 MB·English

by  Popov M., Randrianantoanina B.

De Gruyter Studies in Mathematics 45
Editors
CarstenCarstensen,Berlin,Germany
NicolaFusco,Napoli,Italy
FritzGesztesy,Columbia,Missouri,USA
NielsJacob,Swansea,UnitedKingdom
Karl-HermannNeeb,Erlangen,Germany

Mikhail Popov
Beata Randrianantoanina
Narrow Operators
on Function Spaces
and Vector Lattices
De Gruyter
MathematicsSubjectClassification2010:Primary:46B20;Secondary:46B03,46B10.
ISBN978-3-11-026303-9
e-ISBN978-3-11-026334-3
ISSN0179-0986
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ToMila,AnitaandYola
Preface
Most classes of operators that are not isomorphic embeddings are characterized by
some kind of a“smallness”condition. Narrowoperators arethose operators defined
on function spaces that are “small” at signs, i.e. at ¹(cid:2)1;0;1º-valued functions. The
ideato consider suchoperators has ledto manyinteresting problems thatcanbe ap-
pliedtogeometricfunctionalanalysis,operatortheoryandvectorlattices.
NarrowoperatorswereformallydefinedandnamedbyPlichkoandPopovin1990
(see [110] and [115]) for operators acting from a rearrangement invariant function
F-space with an absolutely continuous norm to an F-space. However, several au-
thors studied this type of operators earlier, including Bourgain [19] (1981), Bour-
gain and Rosenthal [20] (1983), Ghoussoub and Rosenthal [44] (1983), and Rosen-
thal [126, 127, 128] (1981–1984). In [44] the so-called norm-sign-preserving oper-
ators on L wereconsidered, which are exactlythe nonnarrow operators. There are
1
alsotwocitations thathave anessentialinfluence onthe theoryof narrowoperators,
even though they do not explicitly mention narrow operators: the book by Johnson,
Maurey,SchechtmanandTzafriri[49],andTalagrand’spaper[138].
ThefirstsystematicstudyofnarrowoperatorswasconductedbyPlichkoandPopov
inthememoir[110]mentionedabove. In1996V.KadetsandPopov[57]extendedthe
notionofnarrowoperatorstooperatorsonC.K/-spaces.In2001V.Kadets,Shvidkoy
andWerner[63]introducedanothernotionofnarrowoperatorswithdomainsequalto
BanachspaceswiththeDaugavetproperty. In2005V.Kadets,KaltonandWerner[53]
introducedhereditarilynarrowoperators.In2009O.Maslyuchenko,Mykhaylyukand
Popov[93]extendedthedefinitionofnarrowoperatorstooperatorsdefinedonvector
lattices.
This book describes the current theory of narrow operators defined on function
spaces and vector lattices. We aim to give a comprehensive presentation of known
results and to include a complete bibliography. The only topic that we do not de-
scribeindetailaretheoperatorsintroducedbyV.Kadets,ShvidkoyandWerner[63],
which are also called narrow operators, but which are very different from our nar-
row operators. Their theory is basedon a completelydifferentideaandis actuallya
partof the modern theoryof Banachspaceswiththe Daugavetproperty; wesuggest
that this class of operators should be named Daugavet-narrow. These operators are
ofgreatinterest,buttheydeserveamonograph oftheirown,andthereisnotenough
spaceheretopresentallnecessarybackgroundinformationfortheirstudy. Webriefly
mention theminSection11.3without giving anydetails,but doprovide the relevant
bibliography.
viii Preface
Chapter1containspreliminariesonF-spaces,Köthefunctionspaces,operatorthe-
oryandvectorlattices,includingthedefinitionandinitialpropertiesofnarrowopera-
torsdefinedonaKöthefunctionF-spaceonanatomlessmeasurespace.(cid:2);†;(cid:3)/.
InChapter2weshowthattheclassofnarrowoperatorscontainscompactandAM-
compact operators, Dunford–Pettis operators, operators whose ranges have smaller
density than the domain space, and some other classes. We also show that every
narrow operator from E to a Banach space can be restricted to a suitable subspace
isometrically isomorphic to E, in such a way thatthe restrictionis compact and has
anarbitrarilysmallnorm.
It turns out thatfor a large classof strictlynonconvex Köthe function F-spacesE
including L .(cid:3)/ with 0 < p < 1, the only narrow operator defined on E is zero.
p
Usingthisfact,inChapter3weshowthatahomogeneousnonseparableL .(cid:3)/-space,
p
with0 < p < 1, hasno nontrivial separablequotient space;wealsogive anelegant
isomorphicclassificationofaclassofspaces,whichwecallstrictlynonconvexKöthe
functionF-spaces.
Chapter4isdevotedtoanexampleofanarrowprojectionofarearrangementinvari-
ant(r.i.)spaceE onto asubspaceisomorphic toE,showingthattheclassof narrow
operatorsisnotcontainedinanyotherclassof“small”operators,includingcompact
operators and strictlysingular operators. In the separablecase,this projection is de-
scribedastheintegrationoperatorwithrespecttoonevariableactingonfunctionsof
twovariables. Thisoperatoralsoplaysanimportantroleinothercounterexamples.
InChapter5wedealwiththefollowingnaturalquestionsaboutnarrowoperators:
Whatsubsetsofthecomplexplanecouldbespectraofnarrowoperators? Isthecon-
jugate operator of a narrow operator, narrow? Is the sum of two narrow operators
narrow? Does the set of all narrow operators have the right-ideal property? Do nu-
mericalradiiofnarrowoperatorsapproximatethenumericalindexofL ?
p
ItiswellknownthatL hastheDaugavetproperty, thatis,theDaugavetequation
1
kICKkD1CkKkissatisfiedforeveryweaklycompactoperatorKonL whereI
1
istheidentityofL . InChapter6weshowthattheDaugavetpropertyandsomeofits
1
generalizations hold for narrow operators, and present applications to the geometric
structureofL .(cid:3)/-spaces. Inparticular,foreach1(cid:3) p <1,p ¤2,thereisacon-
p
stantk > 1suchthatifX isacomplementedsubspaceofL andtheprojectionP
p p
fromL ontoX satisfieskI (cid:2)Pk < k thenX isisomorphic toL . Further,ifthe
p p p
Banach–MazurdistancebetweentwospacesL .(cid:3) /,i D1;2,islessthatk thenthe
p i p
corresponding measure spaces have isomorphic homogeneous parts are isomorphic,
uptoconstantmultiples,fordetailsseeSection6.4.
WeshowedinChapter4thatnarrownessdoesnotimplystrictsingularity. InChap-
ter 7 westudy in whatsituations various versions of strictsingularity imply narrow-
ness. This chapter contains some of the deepest results of this book. Many of them
were obtained before the notion of narrowness was formally defined. We present
the theorem of Bourgain and Rosenthal [20] that every ` -strictly singular operator
1
Preface ix
fromL toaBanachspaceX is narrowandtheverydeepRosenthal’s characteriza-
1
tionofnarrowoperatorsonL [128],whichinparticularimpliesthateveryL -strictly
1 1
singularoperatoronL ,alsocalledanon-EnflooperatoronL ,isnarrow. Wepresent
1 1
this resulttogether withits connections with pseudo-embeddings, pseudonarrow op-
erators and the Enflo–Starbird maximal function (cid:4). This combines results of Enflo
and Starbird [37], Kalton [66] and Rosenthal [128]. We alsopresent the theorem of
Johnson, Maurey,SchechtmanandTzafriri’sthateveryL -strictlysingular operator
p
onL ,i.e.everynon-EnflooperatoronL ,isnarrow. Wefinishthechapterwithsome
p p
applicationsoftheseresults. Thestudyof` -strictlysingular operatorslogicallybe-
2
longs in this chapter, but the two known partial results require additional techniques
sowepresenttheminChapters9and10,respectively.
In Chapter 8 we discuss different notions of “weak” embeddings of L , namely
1
semi-embeddings,G -embeddingsandsign-embeddings. WealsopresentTalagrand’s
ı
[138]constructionofasubspaceX ofL suchthatL doesnotisomorphicallyembed
1 1
in either X or L =X. This is interesting for us, becausethe corresponding quotient
1
mapisanonnarrowoperatorfromL toaBanachspacethatcontainsnoisomorphic
1
copyofL . Thus,anL -strictlysingularoperatordefinedonL ,thatis,anon-Enflo
1 1 1
operatordefinedonL , doesnothavetobenarrowifthe rangespaceisanarbitrary
1
Banachspace.
Chapter 9 contains all known information concerning the Banach spaces X for
whicheveryoperator from L to X is narrow. Heretwo factsshould be mentioned.
p
Every operator from L to L is narrow if 1 (cid:3) p < 2 and p < r < 1, and
p r
this is no longer true for any other values of p and r. The second resultassertsthat
every operator from L to ` is narrow if r ¤ 2. The techniques developed in this
p r
chapter,whicharequite interestingandinclude aprobabilistic approach, allowusto
prove a partial result concerning narrowness of ` -strictly singular operators, which
2
ispresentedinSection9.5.
Oneofthemoststrikingfactsconcerningnarrowoperatorsisthat,ifanr.i.function
spaceE hasanunconditionalbasistheneveryoperatoronE isasumoftwonarrow
operators. In contrast, the sum of two narrow operators on L is narrow. These
1
phenomena areexplainedthrough the extensionof thenotion of narrowoperators to
vector lattices. O. Maslyuchenko, Mykhaylyuk and Popov [93] (2009) proved that
thesetofallnarrowregularoperators(i.e.differencesofpositiveoperators)between
lattices that are “nice enough,” including L , form a band, and so, in particular a
p
sum of two narrow regular operators is narrow, like for operators on L . In fact all
1
operatorsonL areregular,sothephenomenonofsumsonL isaspecialcaseofthe
1 1
generalbehaviorofregularnarrowoperatorson“nice”vectorlattices. InChapter10
we present a generalization of Kalton’s and Rosenthal’s representation theorems for
operators on L to vector lattices, which was proved in [93]. The last Section 10.9
1
contains a generalization of a result of Flores and Ruiz [39] about narrowness of
regular` -strictlysingularoperators.
2
x Preface
Chapter11containssomevariantsofthenotionofnarrowoperators. Oneofthem,
hereditarily narrow operators, allowed V. Kadets, Kalton and Werner [53] to prove
the strongest generalization of Pełczyn´ski’s theorem on the impossibility of the iso-
morphicembeddingofL intoaBanachspacewithanunconditionalbasis. Another
1
variant, gentle narrow operators introduced in [102], is usedto give a partialanswer
to the problem whether Rosenthal’s characterizationof narrow operators on L can
1
begeneralizedtoL for1 (cid:3) p < 2. NextwepresentthenotionofC-narrowopera-
p
torsonC.K/-spacesdefinedin[57]. SinceC.K/-spacesdonotcontaincharacteristic
functions, the definition of C-narrowoperators is basedon Rosenthal’scharacteriza-
tion of narrow operators on L . This approach proved quite fruitful and C-narrow
1
operators share many properties of narrow operators on Köthe–Banach spaces. The
lasttwosectionsaredevotedtotheusualnotionofnarrowoperatorsbutinsomewhat
unusualsettings. Mostoftheresultsonnarrowoperatorsusetheabsolutecontinuity
ofthenormofthedomain. Investigationofnarrowoperatorsdefinedonspaceswith-
outthisproperty,likeL1,leadstomanysurprisingresults. Forexample,thereexist
compactoperatorsandevenlinearfunctionalsonL1thatarenotnarrow. Wepresent
the known results and open problems in this setting in Section 11.4. We finish the
chapterwitharesultthatevery2-homogeneousscalarpolynomialonL ,1(cid:3)p <2,
p
isnarrow. Wethinkthatitwouldbeinterestingtoinvestigatethenotionofnarrowness
forpolynomialsonBanachspaces.
A number of proofs in this book are new, and some of them are due to our col-
leagues. Wheneverwepresenttheirproofs,wegratefullycredittheauthors.
The concept of narrow operators, a subject of numerous investigations during the
last30years,gaverisetoanumberofattractiveopenproblems. Westatetheseprob-
lemsthroughoutthisbooknearthecontextfromwhichtheyoriginate. Fortheconve-
nienceofthereader,inthelastChapter12welistallopenproblemsthatwerestated
indifferentchapters. Wehopethatthebookwillinspirenewworkontheseproblems.
ThisbookwasstartedduringthevisitofthefirstnamedauthortoMiamiUniversity
inOxford, Ohio,USA,forthe2010/11 academicyear. Hethanks theDepartmentof
MathematicsandMiamiUniversityfortheirhospitalityandfinancialsupport.
We are grateful to our coauthors A. Dorogovtsev, V. Kadets, O. Maslyuchenko,
V.MykhaylyukandA.Plichkofortheircontributionstonewproofsofdifferentresults
presentedinthebook,andtoA.Kusraev,M.OstrovskiiandM.Plievfortheirhelpful
comments on preliminary drafts of the book. We also thank G.J.H.M. Buskes for
bringingtoourattentionthereference[99].
Chernivtsi/Oxford(Ohio), MikhailPopov,
May2012 BeataRandrianantoanina