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Disjunctive Linear Operators and Partial Multiplications in Riesz Spaces

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DISJUNCTIVE LINEAR OPERATORS AND PARTIAL MULTIPLICATIONS IN RIESZ SPACES 0000 Promotor: dr.B . van Rootselaar, hoogleraar in de wiskunde YY/ï Gto \ ^So c B. VAN PUTTEN DISJUNCTIVE LINEAR OPERATORS AND PARTIAL MULTIPLICATIONS IN RIESZSPACE S PROEFSCHRIFT TER VERKRIJGING VAND EGRAA DVA N DOCTOR IND ELANDBOUWWETENSCHAPPEN , OP GEZAGVA ND ERECTO R MAGNIFICUS DR. H.C.VA NDE RPLAS , HOOGLERAAR IND EORGANISCH E SCHEIKUNDE, IN HET OPENBAART EVERDEDIGE N OP WOENSDAG 17DECEMBE R 1980 DES NAMIDDAGST EVIE RUU RI ND EAUL A VAND ELANDBOUWHOGESCHOO LT EWAGENINGEN . Krips Repro- Meppe l Y?/W: \%°>O%LÓ^ Want htt dwaze Goch, ÂJ, W-ijzVi dan do. mznizn (7 KoAÂnthz 1 : 25a) BIBLIOTHEEK LH. 1 6 DEC. n«ITV. T1JDSCHR. AlïM. /tfAJOüïö' ,2*30 STELLINGEN I Iedere semi-normp o pee n eindig dimensionale Archimedische Riesz ruimte E zodanig dat p(x)= p(|x| )voo r allex e Ei see nRies z semi-norm. De verwijzing die Aliprantise nBurkinsha w voor deze stelling geveni s niet terecht omdat deze stelling reedsi n196 1i sbeweze n door Bauer, Stoere nWitzgall . Aliprantis, CD, enBurkinshaw ,0 . Locally solid Riesz spaces, AcademiePres s (1978). Bauer, F.L., Stoer,J .e nWitzgall ,C . Absolute and monotonie norms, Numer. Math.,Vol .3 ,pp . 257-264 (1961). II Alsp ee n reguliere vectoriële pseudonorm: £ ^ IR is,N(p )={ xe £n; p(x)=0 } en R(p)= (p(x);x e %} ,da ni sdi mR(p )< n - di m N(p). III Also pee n Archimedische Riesz algebra Eee n L-norm II-II gedefinieerd is dan bestaate ro pE ee nsemi-inproduct . AlsE ee n Archimedische $-algebrai se n(E ,II .II )ee n AL-ruimte,da ni s er een norm II-II 'o pE zodani gda t(E ,II -II')ee nHuber t ruimteis . IV De lineaire ruimte B(X,Y)me t operatornorm van alle norm begrensde lineaire operatoren vanX naa r Y,waarbi jX * {0}e nY genormeerd e lineaire ruimten over F zijn,i sjuis t dan norm compleet alsY nor m compleet is. Het analogonva ndez e stelling inRies z ruimten,me t orde begrensd inplaat s van norm begrensde nDedekin d compleeti nplaat sva n norm compleet,i sgeldi g onderd ebeperkin g datd eord e dualeX vanX voldoe t aanX =£{0} . Pfaffenberger, W.E. A converse to a completeness theorem, Amer. Math. Monthly, Vol. 87,no .3 ,p .21 6(1980) . Aliprantis, CD. On order properties of order bounded transformations, Canad.J .Math. ,Vol . 27,no .3 ,pp .666-678 . V AlsX ee n lineaire ruimte over K is,Y ee nDedekin d complete Riesz ruin en p:X - >Y eensublineair e operatorva nd evor m |Lx| (Lee nlineair e operator vanX naa r Y),da ni svoo r iederex e X d everzamelin g K ={ ye X ; p(x+ y )= p(x )+ p(y) }ee npre-kege l in X. VI Zij voor allen e H den x n matri x A gegeven door A (i,j)= e ' 1J ' (i,j = 1 ,.. ., n). e+ 1 Voord espectraa lstraa l p(A )va n A geldt lim p(A)= _ •, . VII Vekslere nGeile r hebben bewezenda tieder e voorwaardelijk lateraal com­ plete Archimedische Riesz ruimted eprojecti e eigenschap bezit. Bernau zegtva ndez e stellingee ngeneralisati e tegeve n intralie-geor ­ dende groepen. Integenstellin g tot zijn bewering iszij n resultaat voor Riesz ruimten zwakker dan dat van Vekslere nGeiler . Veksler, A.I.e nGeiler , V.A. Order and disjoint completeness of linear partially ordered spaces (Russisch), Sibirsk Mat. 1., Tom. 13,pp . 43-51, English transi.: Siberian Math.J. ,Vol . 13,pp . 30-35. Bernau, S.J. Lateral and Dedekind completion of archimedean lattice groups, J. London Math. Soc. (2),Vol . 12,pp .320-322 . VIII De statistische selectie-e nrangschikkingstechnieken , waarvoor recent ind estatistisch e literatuur veel belangstelling aand eda g isgelegd , verdienen inhe tlandbouwkundi g onderzoek grote aandacht. IX Het verdient aanbeveling deuitdrukkin g "een x-aantal" niett ebezige n zonder specificatie vanx . X De overheid dientte n aanzienva nalternatiev e groeperingen geen alter­ natieve gedragslijn tevolgen . Stellingen bij het proefschrift "Disjunctive linear operators andparti e multiplications inRies z spaces". B.va nPutte n Wageningen,1 7decembe r 19Î VOORWOORD De aanleiding tot het schrijven van dit proefschrift was een projectvoor­ stel, gericht aan de Vaste Commissie voor de Wetenschapsbeoefening van de Landbouwhogeschool,da t als titel had "vectorial norms on linearspaces" . Ditwa s tevens de titel van mijn afstudeerverslag (mei 1976). Ditproject ­ voorstel werd goedgekeurd en in september 1977wer d begonnen met de uit­ werking van dit project. Daartoe werden normen bestudeerd,di e waarden aannemen in Dedekind comple­ te Riesz ruimten.A l spoedig kreeg ik daarbij temake n met de operatoren, die in dit proefschrift een belangrijke plaats innemen en die zeer de moeitewaar d leken om teworde n bestudeerd. Voor verdere studie van heton ­ derwerp was kennis van deze operatoren noodzakelijk en zo langzamerhand werden de ruimten die het bereik zijn van deze normen het domein van studie. Intussen bleek dat de operatoren die mijn belangstelling hadden ook bestu­ deerd werden door enkele anderen,meesta l vanuit verschillende achtergronden. Het resultaat is nu een proefschrift op het gebied van de zuiverewiskunde ; de titel ervan heeft met de titel van het projectvoorstel toch nog twee woorden gemeen. Bij dezen dank ik mijn ouders voor hun zorg door alle jaren heen en ook mijn overige familie dank ik voor hun belangstelling. Professor Dr.A . van der Sluis dank ik vriendelijk voord e fijne afstudeer- periode en voor het vele dat ik in die tijd van hem heb geleerd. Mijn promotor,Professo r Dr.B . van Rootselaar,dan k ik hartelijk voor zijn hulp en zijn aanmoediging om de sprong in het oneindig dimensionale temaken . Ook de overige leden van de Vakgroep Wiskunde dank ik voor hun belangstel­ ling, in het bijzonder Drs.B.R . Damsté voor het corrigeren van de tekst in de zeer korte tijd die daarvoor nog overig was. Iwoul d like to thank Professor Dr.W.A.J . Luxemburg (Pasadena,California ) and Professor Dr.H.H . Schaefer (Tübingen,B.R.D. ) for their kind invitation to visit the conference about "Riesz spaces and order bounded operators", which was held in Oberwolfach, from June 24 till June 30,1979 . Jodien Houwers en Ans van der Lande-Heij dank ik voor het typen van dit proefschrift. Mijn vrouw Riet dank ik voorhaa r steun en voor het accepteren vanmij n dag­ en avondindeling. AMS(MOS) subject classifications (1980) 06F20,47B55 . CONTENTS page Chapter I. RIESZSPACE S ] 1.Orde r relation 1 2. Partially ordered linear spaces and Riesz spaces 3 3. Elementary properties of Riesz spaces 6 4. Linearoperator s 13 Chapter II. SOMETYPE SO F CONVERGENCE 19 5. Sequences in Riesz spaces 19 6. Order convergence 19 7. Regulator convergence 21 8. Characteristic convergence 23 9. Comparison of convergences of sequences 25 10. Some notions of ideals 27 11. Continuity of linearoperator s 29 Chapter III.DISJUNCTIV E LINEAR OPERATORS 32 12. Orthomorphisms 32 13. Nullspaces of disjunctive linear operators 40 14. Examples of unbounded stabilizers 51 Chapter IV. f-ALGEBRAS 53 15. Introduction to f-algebras 53 16. Faithful f-algebras 57 17. Ideals and ring ideals 60 18. Ring homomorphisms inArchimedea n f-algebras 63 19. Normalizers and local multipliers on f-algebras 69 Chapter V. INVERSIONS AND SQUARE ROOTS INARCHIMEDEA N $-ALGEBRAS 72 20. Inversion inArchimedea n $-algebras 72 21. Square roots inArchimedea n $-algebras 76 22. A generalization ofth e Ellis-Phelps theorem 79 Chapter VI. PARTIAL f-MULTIPLICATION INARCHIMEDEA N RIESZSPACE S 81 23. Introduction and definitions 81 24. Intrinsically defined p<ï>-multipl icatio n onArchimedea n Riesz 88 spaces with a strong order unit 25. Orth(E)fo r Ea n Archimedean Riesz spacewit h a strong order 92 unit REFERENCES 95 SAMENVATTING 100 CURRICULUM VITAE 101 ChapterI RIESZSPACE S Inthi s chapterw e give anexpositio n of theelement s of the theory of Rieszspace ;fo ra shorthistorica l introduction we refer to thebook s ofAlipranti s and Burkinshaw [1978] and Luxemburg and Zaanen [1971]. 1.Orde r relation An order relation on a non-empty set Si sa subset< of the Cartesian product Sx S o fS wit h the following properties: < istransitive ,i.e . if (x,y)e < and (y,z)e < for x,y,ze s,the n also (x,z)e < , < is reflexive,i.e . (x,x)e < for all xe S,< is anti-symmetric,i.e . (x,y)e < and (y,x)e < forx,y' eS implie s x= y . Iti scommo n towrit e x< y instead of (x,y)e < . A partially ordered set is now defined as apai r (S,<)wher e Si sa non-empty setan d< is anorde r relation on S. Twoelement s xan dy are called comparable ifa t least one of thestate ­ ments x< y ,y < xholds ,otherwis e theyar ecalle d incomparable. A non-empty subset Xo fa partially ordered set (S,<) iscalle d achai n in (S,<) ifal l pairs ofelement s of Xar ecomparable . (S,<) is said tob e totally ordered ifS itsel f isa chai n in (S,<). Ifx isa n element ofa partially ordered set (S,<) such that x< y for y s s implies x= y , then x iscalle d amaxima l element of (S,<). x isa minima l element of (S,<) ify < xfo ry eS implie sy =x . Forelement s x,y,z ofa partially ordered set (S,<) the following nota­ tions are used: x< y for x< y &x # y; y > xfo r x< y; y > xfo r x< y; x< y< zfo rx < y &y < z; x,y< z for x< z& y < z; [x,y] for the order interval {zeS; x< z< y} .
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    REMARKS ON THE RIESZ-KANTOROVICH FORMULA D. V. RUTSKY Abstract. The Riesz-Kantorovich formula expresses (under cer- tain assumptions) the supremum of two operators S; T : X ! Y where X and Y are ordered linear spaces as S _ T (x) = sup [S(y) + T (x − y)]: 06y6x We explore some conditions that are sufficient for its validity, which enables us to get extensions of known results characterizing lattice and decomposition properties of certain spaces of order bounded linear operators between X and Y . 0. Introduction Ordered linear spaces, and especially Banach lattices, and various classes of linear operators between them, play an important part in modern functional analysis (see, e. g., [3]) and have significant appli- cations to mathematical economics (see, e. g., [9], [10]). The following famous theorem was established independently by M. Riesz in [19] and L. V. Kantorovich in [16] (all definitions are given in Section 1 below; we quote the statement from [8]). Riesz-Kantorovich Theorem. If L is an ordered vector space with the Riesz Decomposition Property and M is a Dedekind complete vec- tor lattice, then the ordered vector space Lb(L; M) of order bounded linear operators between L and M is a Dedekind complete vector lat- b tice. Moreover, if S; T 2 L (L; M), then for each x 2 L+ we have (1) [S _ T ](x) = supfSy + T (x − y) j 0 6 y 6 xg; (2) [S ^ T ](x) = inffSy + T (x − y) j 0 6 y 6 xg: arXiv:1212.6243v1 [math.FA] 26 Dec 2012 In particular, this theorem implies that under its conditions every order bounded operator T : L ! M is regular.
  • A. G" Kusraev and S. S. Kutateladze Subdifferentials: Theory And

    A. G" Kusraev and S. S. Kutateladze Subdifferentials: Theory And

    Mathematicsand lts Applications A. G" Kusraevand S.S. Kutateladze Subdifferentials: Theoryand Applications KluwerAcademic publishers Subdifferentials: Theory andApp htcations b A. G. Kusraev North,O ssetian Sn rc Ut t i vers i ty, vtadtkavkLz,Russia and S.S. Kutateladze In.uit ute of M athe nnt i cs. ttbertanDivision oIthe Rus.siattAcademy of Sciences, Novosibirsk,Russia \s 7W KLUWERACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON Contents Preface vii Chapter 1. Convex Correspondences and Operators 1. Convex Sets .................................................. 2 2. Convex Correspondences ..................................... 12 3. Convex Operators ............................................ 22 4. Fans and Linear Operators ................................... 35 5. Systems of Convex Objects ................................... 47 6. Comments ................................................... 58 Chapter 2. Geometry of Subdifferentials 1. The Canonical Operator Method ............................. 62 2. The Extremal Structure of Subdifferentials ................... 78 3. Subdifferentials of Operators Acting in Modules .............. 92 4. The Intrinsic Structure of Subdifferentials .................... 108 5. Caps and Faces .............................................. 123 6. Comments ................................................... 134 Chapter 3. Convexity and Openness 1. Openness of Convex Correspondences ......................... 137 2. The Method of General Position .............................. 151 3. Calculus of Polars ...........................................