ON SOME CONSEQUENCES OF THE ISOMORPHIC
ON SOME CONSEQUENCES OF THE ISOMORPHIC CLASSIFICATION OFCARTESIAN PRODUCTS OF LOCALLY CONVEX SPACESA THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCESOFMIDDLE EAST TECHNICAL UNIVERSITYBYERSIN KIZGUTIN PARTIAL FULFILLMENT OF THE REQUIREMENTSFORTHE DEGREE OF DOCTOR OF PHILOSOPHYINMATHEMATICSJULY 2016
Approval of the thesis:ON SOME CONSEQUENCES OF THE ISOMORPHIC CLASSIFICATION OFCARTESIAN PRODUCTS OF LOCALLY CONVEX SPACESsubmitted by ERSIN KIZGUT in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Mathematics Department, Middle East TechnicalUniversity by,Prof. Dr. M. Gülbin Dural ÜnverDean, Graduate School of Natural and Applied SciencesProf. Dr. Mustafa KorkmazHead of Department, MathematicsProf. Dr. Murat YurdakulSupervisor, Mathematics Department, METUExamining Committee Members:Prof. Dr. Mefharet KocatepeMathematics Department, Bilkent UniversityProf. Dr. Murat YurdakulMathematics Department, METUProf. Dr. Zafer NurluMathematics Department, METUProf. Dr. Eduard Emel’yanovMathematics Department, METUProf. Dr. Mert ÇağlarMathematics and Computer Science Department, IKUDate:
I hereby declare that all information in this document has been obtained andpresented in accordance with academic rules and ethical conduct. I also declarethat, as required by these rules and conduct, I have fully cited and referenced allmaterial and results that are not original to this work.Name, Last Name:Signatureiv:ERSIN KIZGUT
ABSTRACTON SOME CONSEQUENCES OF THE ISOMORPHIC CLASSIFICATION OFCARTESIAN PRODUCTS OF LOCALLY CONVEX SPACESKızgut, ErsinPh.D., Department of MathematicsSupervisor: Prof. Dr. Murat YurdakulJuly 2016, 36 pagesThis thesis takes its motivation from the theory of isomorphic classification of Cartesian products of locally convex spaces which was introduced by V. P. Zahariuta in1973. In the case X1 X2 Y1 Y2 for locally convex spaces Xi and Yi , i 1, 2;it is proved that if X1 , Y2 and Y1 , X2 are in compact relation in operator sense, itis possible to say that the respective factors of the Cartesian products are also isomorphic, up to their some finite dimensional subspaces. Zahariuta’s theory has beencomprehensively studied for special classes of locally convex spaces, especially forfinite and infinite type power series spaces under a weaker operator relation, namelystrictly singular. In this work we give several sufficient conditions for such operatorrelations, and give a complete characterization in a particular case. We also show thata locally convex space property, called the smallness up to a complemented Banachsubspace property, whose definition is one of the consequences of isomorphic classification theory, passes to topological tensor products when the first factor is nuclear.Another result is about Fréchet spaces when there exists a factorized unbounded operator between them. We show that such a triple of Fréchet spaces (X, Z, Y ) has acommon nuclear Köthe subspace if the range space has a property called (y) whichwas defined by Önal and Terzioğlu in 1990.v
Keywords: Isomorphic classification of Cartesian products, unbounded operators,strictly singular operators, compact operators, smallness up to a complemented Banach subspace propertyvi
ÖZYEREL KONVEKS UZAYLARIN KARTEZYEN ÇARPIMLARININİZOMORFİK SINIFLANDIRILMASININ BAZI SONUÇLARI ÜZERİNEKızgut, ErsinDoktora, Matematik BölümüTez Yöneticisi: Prof. Dr. Murat YurdakulTemmuz 2016 , 36 sayfaBu tez motivasyonunu V. P. Zahariuta tarafından öncülük edilen yerel konveks uzayların Kartezyen çarpımlarının izomorfik sınıflandırılması teorisinden almaktadır. Xive Yi , i 1, 2 yerel konveks uzayları verilmiş olsun. X1 X2 Y1 Y2 durumundaçarpan uzayların da sonlu boyutlu birer altuzay hariç izomorfik olabilmesi için X1 , Y2ve Y1 , X2 uzayları arasında operatör teorisi bağlamında bir kompakt bağıntı olmasıgerektiği ispatlanmıştır. Zahariuta’nın bu teorisi daha sonra sonlu ve sonsuz tipi kuvvet toplamlı uzaylar başta olmak üzere bazı özel yerel konveks uzaylar için dahazayıf operatör bağıntıları-strictly singular-altında detaylı bir şekilde ele alınmıştır. Buçalışmada söz konusu operatör bağıntılarının varlığı için yeterli koşullar türetilmişve belli bir durumda karakterizasyon elde edilmiştir. Bunun dışında izomorfik sınıflandırma teorisinin sonuçlarından biri olarak yerel konveks uzaylar için tanımlananSCBS (tümlenebilen bir Banach altuzayı dışında yeterince küçük olma) özelliğinin,ilk çarpanın nükleer olması koşuluyla, topolojik tensör çarpımına geçtiği ispatlanmıştır. Bir diğer sonuç ise Fréchet uzayları üzerine olup, iki Fréchet uzayı arasındatanımlı üçüncü bir Fréchet uzayı üzerinden çarpanlarına ayrılan bir sınırsız operatörünvarlığına dayanmaktadır. Bu durumun sonucunda bu üç uzayın ortak nükleer Köthealtuzayı olabilmesi için, görüntü uzayında Önal ve Terzioğlu tarafından 1990’da tanımlanan (y) özelliğinin olmasının yeterli olduğu ispatlanmıştır.vii
Anahtar Kelimeler: Kartezyen çarpımların izomorfik sınıflandırılması, sınırsız operatörler, strictly singular operatörler, kompakt operatörler, SCBS özelliğiviii
To the memory ofProf. Dr. Tosun Terzioğluix
ACKNOWLEDGMENTSI would like to thank to my supervisor Prof. Dr. Murat Yurdakul for his encouragement and his constant support. I am grateful to Prof. Dr. Mefharet Kocatepe and Prof.Dr. Zafer Nurlu for their contributions and corrections during the progress meetings.I will remember Prof. Dr. Eduard Emel’yanov and Prof. Dr. Mert Çağlar with theirfriendly approaches and motivating comments. I also thank to Prof. Dr. Joseph A.Cima for drawing my attention to parallel developments, and related open problemsin complex analysis.It is an honor for me to acknowledge the faculty at METU Mathematics Department,especially Prof. Dr. Bülent Karasözen and Dr. Muhiddin Uğuz, for every technical,intellectual or visionary contribution they have given.I acknowledge the Scientific and Technological Research Council of Turkey for supporting this thesis work partially with the program BIDEB 2211.I would not forget to thank to Dr. Emre Sermutlu who inspired me as a scientist.I owe my gratitudes to Anıl Tarar, for not only being a lifelong foul-weather friendbut also a mentor to me. I thank to the colleagues at METU Mathematics Departmentespecially my close friends Dr. Murat Uzunca and Ayşegül Kıvılcım for their sincerityand supportiveness. I also thank to my colleague Elif Uyanık for sharing her ideasupon reading the manuscript, and for our cooperation in each step during the graduateschool.My parents Leyla Kızgut, Dr. İsa Kızgut; and my wife Başak Kızgut deserve muchmore than my special thanks for their love, respect, patience and understanding duringthis graduate work of which I hope the result is worthy.x
TABLE OF CONTENTSABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xTABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiLIST OF NOTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiCHAPTERS1INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12OPERATORS IN LOCALLY CONVEX SPACES . . . . . . . . . . . .32.1Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . .32.2Compact operators . . . . . . . . . . . . . . . . . . . . . . . . .52.3Strictly singular operators . . . . . . . . . . . . . . . . . . . . .62.4Factorized unbounded operators . . . . . . . . . . . . . . . . . 103ISOMORPHIC CLASSIFICATION OF CARTESIAN PRODUCTSOF LOCALLY CONVEX SPACES . . . . . . . . . . . . . . . . . . . . 133.1Strictly singular operators and isomorphic classification . . . 133.2Sufficient conditions for (X, Y ) S . . . . . . . . . . . . . . . 143.3Bounded operators and isomorphic classification . . . . . . . 23xi
4TOPOLOGICAL TENSOR PRODUCTS OF LOCALLY CONVEXSPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1The SCBS property . . . . . . . . . . . . . . . . . . . . . . . . 254.2Topological tensor products of Fréchet spaces with SCBSproperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 -Köthe spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 27REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29CURRICULUM VITAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35xii
LIST OF NOTATIONSAbbreviationsICCPThe isomorphic classificaion of Cartesian productstvsTopological vector spacelcsLocally convex spacepssPower series spacewscWeakly sequentially complete (Banach space)SPSchur propertyDPPDunford-Pettis propertyGeneralU (X)The base of absolutely convex closed neighborhoods of thetopological vector space X.B(X)The class of bounded subsets of the space X.X YThere exists a topological isomorphism between the topological vector spaces (X, τ1 ) and (Y, τ2 ).θ(X)The origin of the topological vector space X.V UV is a (infinite dimensional closed) subspace of U .L(X, Y )The set of linear continuous operators defined on X into Y .K(X, Y )The set of compact operators defined on X into Y .W(X, Y )The set of weakly compact operators defined on X into Y .V(X, Y )The set of completely continuous operators from X into Y .S(X, Y )The set of strictly singular operators defined on X into Y .B(X, Y )The set of bounded operators defined on X into Y .(X, Z, Y ) BFThe triple (X, Z, Y ) has bounded factorization property.PA class of Banach spaces satisfying a property P .s(P)The class of locally convex spaces with local Banach spaceseach of which belongs to P.xiii
s(P )The class of locally convex spaces with local Banach spaceseach of which having no infinite dimensional subspaces belonging to P.ωThe set of all scalar sequences.acx(A)Absolutely convex closed hull of the set A.co(A)Closed hull of the set A.xiv
CHAPTER 1INTRODUCTIONThe set of results obtained in this thesis is in connection with the theory of isomorphicclassification of Cartesian products (ICCP) of locally convex spaces (lcs) which wasinitiated by the remarkable note of Zahariuta [70] published in 1973. In that paper hedefined and characterized a relation between locally convex spaces X and Y called therelation K which means that every continuous linear operator T X Y is compact.It is proved that for lcs’s X X1 X2 and Y Y1 Y2 with (X1 , Y2 ) K and(Y1 , X2 ) K being isomorphic to each other is equivalent to the case that the factorsare near isomorphic, that is, they are isomorphic up to their some finite dimensionalsubspaces. Namely, he made use of Fredholm operator theory to compose an ICCPof locally convex spaces. In Chapter 2, we introduce some results concerning therelation K in the class of Banach spaces. We then briefly mention strictly singularoperators on lcs’s in the context of operator ideals and emphasize the situation whentheir class is an operator ideal. The last part of Chapter 2 is devoted to unboundedoperators and their factorization in Fréchet spaces. We prove that the existence of anunbounded operator T X Y over a third Fréchet space Z causes the existence ofa common nuclear Köthe subspace of the triple (X, Z, Y ) when the range space hasthe property (y), which was introduced by Önal and Terzioğlu [60].In 1998, Djakov, Önal, Terzioğlu and Yurdakul [20] investigated the ICCP of a specialclass of lcs’s, called finite and infinite type power series spaces (pss). They modifiedZahariuta’s method to obtain a similar ICCP of pss’s with the help of a weaker operator theoretic relation. This relation is based on the type of operators called strictlysingular. We denote (X, Y ) S iff every operator T X Y is strictly singular. In1
Chapter 3, we give sufficient conditions to obtain this relation under some conditions.First we introduce such conditions in terms of Banach spaces, and then we extendsome of them to the class of lcs’s via projective limit topologies and Grothendieckspace ideals. It is even possible to claim a characterization when we slightly modify the assumptions for (X, Y ) K. These results are helpful to extend the ICCP ofpss’s to general lcs’s. In Chapter 3, we also revisit the advances in the ICCP of lcs’s.We see the consequences of changing the assumptions on various types of spaces inZahariuta’s theorem. Here we also mention about bounded operators. It is denoted(X, Y ) B when every operator between X and Y is bounded. Referring to thenote of Djakov, Terzioğlu, Yurdakul, and Zahariuta [19] which investigate the ICCPof Fréchet spaces up to basic Banach subspaces under the assumption of (X, Y ) B,we finally setup the basis of the definition of smallness up to a complemented Banachsubspace property (SCBS) which is enjoyed by all Köthe spaces. In Chapter 4, wegive its definition and prove that it is stable under topological tensor products, provided that the first factor is nuclear. We also mention the class of -Köthe spaces asa type of generalized Köthe spaces, in which the canonical basis {en } is an unconditional one. Their topological tensor products are not known explicitly. However, withthe help our result, we deduce that this product has the SCBS property when the firstfactor is nuclear.2
CHAPTER 2OPERATORS IN LOCALLY CONVEX SPACES2.1PreliminariesIn this chapter, we focus on the operator theory of lcs’s. Our concentration will be oncompact, strictly singular, bounded and unbounded operators and their roles in composing relations between lcs’s. We give results concerning compact and unboundedoperators. Results on compact operators will be in terms of Banach spaces whichare in particular locally convex. These results rest on weak and strong convergence,hereditary properties, some other well-known vector space properties such as Schurproperty (SP), Dunford-Pettis property (DPP), approximation property and so on.The duality theory of Banach spaces is also used in the proofs. We will continueour discussion of Banach spaces in Chapter 3 in strictly singular operators perspective. We then extend some of these results to the general class by means of projectivelimit topologies, and Grothendieck space ideals. The results for unbounded operatorshighly depend on failure of bounded factorization property and continuous norm arguments. There we consider the class of Fréchet spaces. Now let us define the toolswe need for the proofs.A vector space X over the field K is said to be a topological vector space (tvs) denoted(X, τ ) if X is equipped with the Hausdorff topology τ which is compatible with itsvector space structure (the maps X X X and K X X are continuous).A tvs (X, τ ) is said to be locally convex if it has a base of neighborhoods U {Uα }of the origin consisting of convex sets Uα . Let such U be a filter-base of absolutelyconvex absorbent (a subset S is called absorbent if for all x X there exists r R3
such that for all α K, α r implies x αS) subsets Uα of a vector space Xwith Uα θ. If each set ρUα U for ρ 0 when Uα U , then a lcs (X, τ )αis defined by considering U as a base of neighborhoods of the origin. Each lcs canbe constructed this way. Alternatively, let {pα (x)} be a system of semi-norms on avector space X, such that for each x0 θ there is at least one pα with pα (x0 ) 0.nIf Uα {x X pα (x) 1}, then the system ρU (ρ 0) of U Uαι . Thisι 1base is composed of absolutely convex open sets. Each lcs can also be constructedthis way. A complete metrizable lcs is called a Fréchet space. The metrizable lcs(X, τ ) can always be topologized by a system of absolutely convex neighborhoodsof θ(X). This system constitutes a decreasing sequence. The latter is equivalent tothe topology generated by the increasing sequence of semi-norms associated to theseneighborhoods. A vector space X is called a normed space if its topology is given bya norm, which is a functional satisfying norm axioms. A complete normed space iscalled a Banach space. For a more detailed description, the reader is referred to [38].Let U be an absolutely convex closed neighborhood of a lcs X. N (U ) p 1U (0) is aclosed subspace of X, where pU is the gauge functional of U . Let XU X/p 1U (0)be the quotient space with the norm induced by pU ( ). Its dual is the Banach space X ′ [U ] nU (cf. Definition 2.4.2) with the norm defined by U . If V ρUn 1for some ρ 0 and V U (X) also, then N (U ) N (V ). Let πU X XU be thecanonical quotient map. Then for all U , one can find V U such that there existsφU V XV XU making the following diagram commutative.X IXπV/̃VXφU V/X πŨUXIf φU V A(XV , XU ), for a pre-ideal A of operators then X is called a Grothendieck̃U and X̃V are Banach spaces.A-space and we denote X Groth(A). Here, XThis construction may be useful to get a grip on the nuclearity assumption in Theorem 4.2.1. A nuclear space X actually belongs to Groth(N), where N denotes theclass of nuclear operators. Almost every class of well-known lcs’s is generated by anoperator ideal. A non-example is the class of Montel spaces. For necessary conditions for a class of Hausdorff lcs’s to be generated by an operator ideal one may read[9, Theorem 1].4
Given a family {φk X Xk } of linear maps from a tvs X to tvs’s Xk , the projectivetopology induced on X by the family is the weakest topology on X which makeseach of the maps φk continuous. A family {Xk , φkm } where k, m belong to a directedset I, Xk is a tvs for each k I, {φkm Xm Xk } is a continuous linear map foreach pair k, m I with k m and φkm φmn φkn whenever k m n is called aninverse directed system of tvs’s. The projective limit lim Xk of such a system is the Ðsubspace of the Cartesian product Xk consisting of elements {xk } which satisfyφkm (xm ) xk ,k m. The projective limit lim Xk is a closed subspace of Xk Ðand has the projective topology induced by the family of maps {φk lim Xk Xk } Ðwhere φk is the inclusion lim Xk Xk followed by the projection on Xk . If X is a Ðlcs, and {Xk } are the local Banach spaces for k I then the canonical mappings areφn X Xn , for n I and φnm Xn Xm , for n m.Throughout, unless otherwise stated, a "subspace" always means an infinite dimensional closed subspace, and will be denoted Y X.2.2Compact operatorsDefinition 2.2.1 Let X and Y be lcs’s. T L(X, Y ) is called (weakly) compactif there exists a zero neighborhood U of X such that its image T (U ) is (weakly)precompact in Y .As usual we denote (X, Y ) K (resp. (X, Y ) W) when any operator from Xto Y is compact (resp. weakly compact). Zahariuta [70, Proposition 1.1] characterized (X, Y ) K in the following sense: Y is a pre-Montel (each bounded subset Bof Y is pre-compact) lcs iff (X, Y ) K for each normed space X. Obviously wehave K(X, Y ) W(X, Y ). The converse is not true in general, unless the domainspace has the Schur property (the equivalence of weak and strong convergence) [39,Lemma 9]. Note that (weakly) compact operators have the conjugacy property onBanach spaces, that is, T ′ Y ′ X ′ is (weakly) compact iff T X Y is (weakly)compact. In this section, we introduce some results concerning sufficient conditionsfor (X, Y ) K, where X and Y are Banach spaces. For similar results in the categoryof lcs’s, the reader is referred to [3], [14], [15], [48], and [70, Section III]. To work5
with Banach spaces, we will need some concepts related to sequences and equivalence of their convergence with respect to different topologies. A Banach space Xis said to be weakly sequentially complete (wsc) if weakly Cauchy sequences in Xconverges weakly. It is called almost reflexive if every bounded sequence (xn ) Xhas a weakly Cauchy
a locally convex space property, called the smallness up to a complemented Banach subspace property, whose definition is one of the consequences of isomorphic classi- fication theory, passes to topological
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