MAZUR-ORLICZ THEOREM IN CONCRETE SPACES AND INVERSE PROBLEMS RELATED TO THE MOMENT PROBLEM

Transcription

1 U.P.B. Sci. Bull. Series Vol. 79 Iss. 3 7 ISSN 3-77 ZUR-ORLICZ THEORE IN CONCRETE SPCES ND INVERSE PROBLES RELTED TO THE OENT PROBLE Ocav OLTENU I he firs ar of his wor we derive soe ew alicaios of a versio of azur-orlicz heore i cocree saces of absoluel iegrable fucio ad resecivel coiuous fucios of several real variables. The secod ar is devoed o iverse robles relaed o he arov oe roble. geoeric aroach of aroiaig he soluios of a sse wih ifiiel a euaios ivolvig rascede fucios wih ifiiel a uows is briefl discussed. Kewords: azur-orlicz heore L saces self adoi oeraors arov oe roble iverse robles aheaics Subec Classificaio: Q5 85. Iroducio The versio of azur - Orlicz heore ha we have i id i his wor gives a ecessar ad sufficie codiio for he eisece of a liear osiive oeraor F fro a order vecor sace X io a order colee vecor laice Y such ha F( ) J F() P() X where { } J X {} J Y are give failies P: X Y beig a subliear oeraor []. The relaio F() P() X usuall corols he or of he soluio F. Rece resuls o his subec have bee ublished i [] ad have bee subied i [4]. The firs ai of his wor is o rove soe ew alicaio of azur Orlicz heore o cocree saces X ael o X = L <. The secod urose of his wor is o solve a iverse roble relaed o a arov oe roble (see he bsrac). Fro his oi of view oe coiues he sud sared i [3] [3]. eisece resul for he soluio of a arov oe roble [] is alied. For siilar eisece robles based o Hah Baach heore ad is geeralizaios see [] - [9] [3]. For oeraor valued oe robles see [9] []. For he cosrucio of soe soluios see [9] [3] [3]. The urose of he secod ar of his wor is o aroiae he soluio of a sse wih ifiiel a euaios ivolvig Prof. De.of aheaics Uiversi POLITEHNIC of Buchares Roaia e-ail: oleauocav@ahoo.ie

2 5 Ocav Oleau rascede fucios wih ifiiel a uows sarig fro he oes of a soluio of a arov oe roble. Our soluio is o uiue. This is aoher wa of solvig siilar robles o hose reaed i lieraure b soe oher ehods [4]. Recall ha aoher iora roble i he heor of oes is he uiueess of he soluio [5] - [8]. The bacgroud of his wor is coaied i soe chaers fro [9] []. The res of he aer is orgaized as follows. Secio is devoed o soe alicaios of azur Orlicz heore. I Secio 3 iverse robles relaed o he arov oe roble are discussed. The coclusios are eioed i Secio 4.. licaios of azur Orlicz heore We sar b recallig he geeral absrac for of azur Orlicz heore i he order vecor saces seig. Theore.. (Theore 5 []). Le X be a ordered vecor sace Y a order colee vecor laice { } { J } J arbirar failies i X resecivel i Y ad P: X Y a subliear oeraor. The followig saees are euivale (a) F L(X Y) such ha F( ) J F() X + F() P() X; (b) for a fiie subse J J ad a {λ } J R λ J we have λ X λ P(). J Soe ew alicaios of his geeral resul are deduced i he seuel. Theore.. Le X be a Baach laice Y a order colee Baach laice {φ } J X + { } J Y G a liear osiive bouded oeraor fro X io Y α a osiive uber. The followig saees are euivale (a) here eiss a liear osiive bouded oeraor F B + (X Y) such ha F(φ ) J F() αg( ) X F α G ; (b) αg(φ ) J. Proof. (a) (b) is obvious because of F(φ ) αg( φ ) = αg(φ ) J. For he coverse we al Theore. (b) (a). Le J J be a fiie subse {λ } J R + X such ha φ. The usig (b) ad he fac ha J λ J he scalars λ are oegaive as well as he osiivi of G we derive

3 azur-orlicz heore i cocree saces ad iverse robles relaed o he oe roble 53 λ α J λ G(φ ) = αg ( λ φ αg() αg( ) =: P(). J J ) licaio of Theore. leads o he eisece of a liear osiive oeraor F fro X io Y such ha F(φ ) J F() αg( ) X. Fro he las relaio also usig he fac ha he ors o X ad Y are solid ( u v u v ) we deduce F() αg( ) F() α G = α G X. I follows ha F α G. This cocludes he roof. Corollar.. Le be a easure sace μ a osiive easure o μ() < X = L μ () < g a elee of L μ () where ( ] is he cougae of ( + = ) α a osiive uber. Le {φ } J { } J be as i Theore. where Y = R. The followig saees are euivale (a) here eiss h L μ () h αg a.e. hφ dμ J; (b) α gφ dμ J. Proof. Oe alies Theore. for G(ψ) = g ψdμ ψ X Y = R as well as he rereseaio of liear osiive coiuous fucioals o L saces b eas of oegaive elees fro L saces. I order o rove (b) (a) fro he recedig resuls i follows ha here eiss h (L μ ()) + such ha J ad hψdμ α gψdμ hφ dμ for all oegaive fucios ψ L μ (). Now we choose ψ = χ B where B is a arbirar easurable subse of. The he las relaio ca be rewrie as (h αg) dμ B for all such subses B. sraighforward alicaio of Theore.4 [] leads o h αg a.e. i. Sice (a) (b) is obvious his cocludes he roof. Corollar.. Le cosider he easure sace = R + { } edowed wih he easure dμ = e( = )d d > { } α a osiive uber. The followig saees are euivale

4 54 Ocav Oleau (a) here eiss h L μ (R + ) J! J! R + h dμ N h α a.e.; (b) α + + = ( ) N. Proof. Oe alies Corollar. o = = g = a.e. The oaio is he uli ide oaio =. The coclusio follows via Fubii s heore ad Gaa fucio roeries. Theore.3. Le X = L μ () < μ μ() < {φ } J X { } J R α > α R he cougae of. Cosider he followig saees (a) here eiss h (L μ ()) + such ha hφ dμ J hψdμ α ψ (μ()) ψ X; (b) we have α φ dμ J. The (b) (a). Proof. Le J J be a fiie subse {λ } J R +. Hölder ieuali ad usig also (b) lead o he followig ilicaios λ φ ψ ( λ φ J J ) dμ ψdμ ψ (μ()) λ α ( λ φ ) dμ α J J ψ (μ()) =: P(ψ). licaio of Theore. ad easure heor argues [ heore ] ield he eisece of h L μ () such ha F(φ ) = hφ dμ oreover sice F(ψ) ψ X + we have J F(ψ) = hψdμ α ψ (μ()) hψdμ ψ X +. ψ X. Taig ψ = χ B where B is a easurable se such ha μ(b) > oe obais

5 azur-orlicz heore i cocree saces ad iverse robles relaed o he oe roble 55 hdμ B for all such subses B. licaio of heore.4 [] leads o h μ a. e. Fro he revious relaios we also derive ha h α(μ()). This cocludes he roof. The followig heore rereses a alicaio of he geeral resul saed i Theore. o soe oher cocree saces X Y. Le H be a arbirar Hilber sace N osiive couig self - adoi oeraors acig o H (B ) N a seuece i Y where Y = Y( ) is defied b Y {U (H); U = U = } Y {V Y ; UV = VU U Y }. Here (H) is he real vcor sace of all self adoi oeraors. Oe ca rove ha Y is a order colee Baach laice wih resec o he usual srucures iduced b hose defied o he real sace of self adoi oeraors (see [ ]) ad a couaive real Baach algebra. Noice ha he roeries of Y = Y( ) where are as eioed above ca be roved i a siilar wa o hose of a Y() where is a self adoi oeraor. cuall oe reeas he roofs fro [ ] bu for several couig self adoi oeraors. The Y edowed wih he usual order relaio o selfadoi oeraors is a order-colee vecor laice ad a couaive real Baach algebra [9]. Le also be B he C -algebra geeraed b = ( ) ha is B is he closure i B(H) of eressios of he for P( ) = a J a C = ( ). J N J fiie We a uiuel cosruc he oi secral easure E of he couig = ( ) i B. s i is ow he oi secral easure E is coceraed o he oi secru Σ {γ( ) ; γ( ); γ Γ} σ B ( ) σ B ( ) R Γ {γ: B C; γ is a characer} Because for a se σ Bor(Σ ) we have E (σ) i = E i = i i resuls ha E (σ) Y. Coseuel we have E : Bor(Σ ) (H). The secral easure E = E ( ): Bor(Σ ) (H) is such ha for a oloial = ( ) ( ) Σ of real variables we have ( ) de ( ) = ( ) Σ

6 56 Ocav Oleau Le deoe b φ N he basic oloials φ ( ) = = ( ) N = ( ) Σ.. Theore.4. The followig saees are euivale (a) here eiss a liear bouded osiive oeraor F B + (X Y) such ha F(φ ) B N F(φ) φ de ( ) φ X F ; Σ (b) B = ( ) N. Proof. The ilicaio (a) (b) is obvious: B F(φ ) φ de ( ) = φ de ( ) = Σ Σ N (we have used he osiivi of he oeraors which leads o φ = φ o Σ ). For he coverse oe alies Theore. (b) (a) where N sads for J φ sads for ad B sads for N. Le J ad {λ } J be as eioed a oi (b) of Theore.. The followig ilicaios hold: λ φ J φ X λ φ de ( ) = λ J Σ J φde ( ) φ de ( ) =: P(φ). Σ Σ The osiivi of he secral easure de ( ) has bee used. O he oher had he hohesis (b) he fac ha he scalars λ are oegaive ad he recedig evaluaio ield λ B λ λ B λ J J = λ P(φ) J where P(φ) was defied above. Thus he ilicaio a (b) Theore. is accolished. licaio of he laer heore leads o he eisece of a feasible soluio F havig he roer eioed a oi (a) of he rese heore. The las roer is a coseuece of he recedig oe usig he fac ha he or o Y is solid. This cocludes he roof.

7 azur-orlicz heore i cocree saces ad iverse robles relaed o he oe roble 57 Rear. If i Theore.4. oe addiioall assues ha < = he for a selfadoi oeraors saisfig (b) oe has B (I ). N = 3. iverse roble relaed o a arov oe roble We sar his secio b recallig he followig saee o he absrac oe roble which will be alied i he seuel. Theore 3.. (Theore 4 []). Le X be a ordered vecor sace Y a order colee vecor laice X Y give failies ad J F LX Y (a) here eiss a liear oeraor F LX Y F wo liear oeraors. The followig saees are euivale: such ha F J F F X F ; J (b)for a fiie subse J J ad a R we have: J X F J J F. Theore 3.. Le d ( l ) d l d. S.. ssue ha for a fiie subse S S ad a ; S R The here eiss a uiue h L h a. e. K we have... such ha l ) l h N... ; d d

8 58 Ocav Oleau Proof. Deoe We are goig o al heore 3. o : = φ N. sraighforward couaio ields dν = ( l )d = ( + ). Ne we verif he ilicaio saed a oi (b) Theore 3. where φ sads for J J N. Usig he hohesis ad he above couaio we ifer ha L J J J J d d d F F F :. : Hece he ilicaio fro (b) Theore 3. is accolished. licaio of he laer heore leads o he eisece of a liear fucioal F o X L such ha d. F X The fucioal F has a rereseaio b eas of a fucio h ha has all he roeries eioed a oi (a) b easure heor argues []. This is a coseuece of he rereseaio heore for liear coiuous fucioals o L ν saces wih σ fiie easure (here he easure is fiie) as elees h fro L ν. The ae i he las relaios ψ = χ B (he characerisic fucio of a arbirar ν easurable subse B ν(b) > ). Oe obais h d ν dν = ν(b) B B for all such easurable subses B. sraighforward alicaio of Theore.4 [] leads o he coclusio h a. e. O he oher had he oe

9 azur-orlicz heore i cocree saces ad iverse robles relaed o he oe roble 59 codiios F(φ ) = N of heore 3. (a) (where φ sads for ad sads for ) ield: = F(φ ) = Cosider he sse of euaios h d hd l c.... hφ dν N. l The recedig heore 3. suggess he followig algorih i solvig he sse of euaios (). Se. ssue ha he oes verif codiio (b) of Theore 3.. Fid a aroiaio of he soluio h i ers of he oes N. To his ed sice h L L Hilber base h has a Fourier easio wih resec o he associaed followig Gra-Schid rocedure o he colee sse of liearl ideede oloials. coefficies h are give b: The Fourier h l h l l l l l where l are give b he Gra-Schid rocedure so ha we ow h i ers of he oes. Recall ha here eiss a subseuece of he seuece of Fourier arial sus which coverges oiwise o h. This fac is a coseuece of he rear ha he arial sus of he Fourier series coverge i a L sace. The oe alies heore 3. []. 65. I he seuel we ca wrie: h h where h is a arial su of he Fourier series of h. Noe ha all hese arial sus are oloials so ha he are coiuous. Se. Le h be a arial su of he Fourier series wih resec o he orhogoal oloials. Usig Schwarz ieuali ad aroiaio of coiuous fucios h b sile fucios: h... c... D

10 6 Ocav Oleau The ubers c are he values of h a soe ois i... ;... h where is large ad is suiable chose for aroiaig. h Oe deduces... ) [ ) [ N D d c d c hd where D are oe subses aroiaig i easure he subses h... ;... ad whose cell decoosiios a be wrie as below.... ;... ) [ ) [ h D The above argues ield. l l l l l N N c c d d c

11 azur-orlicz heore i cocree saces ad iverse robles relaed o he oe roble 6 For he oe-diesioal case see [3] Rear 9. The coclusio is ha we ca deeriae (aroiae) he uows... b eas of he cell decoosiio of he oe subses D associaed o he ow oloial h (cf. [ secio.9]). The basic relaios ca be suarized as he sse of euaios () where are give c are ow fro Se ad he uows are deeried i ers of he cell - decoosiio of he suiable chose oe subses D deduced fro he ow oloial h (he easure ν is ouer regular []). The uows are he coordiaes of he verices of he cells (see. [ secio.9]). Clearl he soluio is o uiue. So usig he above oaios we have roved he followig heore. Theore 3.3. aroiaio for he soluio of () is give b he coordiaes { } of he verices of he cells fro he cell decoosiio of he oe subses D associaed o he ow oloials h. For a siilar oe diesioal roble havig a fiie uber of uows ad solved b usig oher ehods see [4]. 4. Coclusios New characerizaios for he eisece of soluios of absrac ad cocree azur Orlicz robles are roved. I he secod ar of his wor a geoeric ehod of aroiaig he soluio of a sse wih ifiiel a euaios ad uows is seched. This is a geeral ehod. Siilar robles ca be solved usig he sae ideas ad aroriae odificaios. Oe uses a differe ehod for relaed robles o hose solved i he lieraure b soe oher ehods. R E F E R E N C E S [] O. Oleau licaio de héorèes de rologee d oéraeurs liéaires au roblèe des oes e à ue gééralisaio d u héorèe de azur-orlicz C. R. cad. Sci Paris 33 Série I (99) [] O. Oleau ad. Oleau licaios of eesio heores of liear oeraors o azur- Orlicz ad oe robles. U. P. B. Sci. Bull. Series 76 3 (4) [3] O. Oleau roiaio arov oe roble ad relaed iverse robles U. P. B. Sci. Bull. Series 78 (6) [4] O. Oleau ad J.. ihăilă Oeraor - valued azur Orlicz ad oe robles i saces of aalic fucios U. P. B. Sci. Bull. Series (acceed). [5] N. I. hiezer The Classical oe Proble ad soe relaed Quesios i alsis Oliver ad Bod Ediburgh-Lodo 965.

12 6 Ocav Oleau [6] C. Berg J. P. R. Chrisese ad C. U. Jese rear o he ulidiesioal oe roble aheaische ale 43 (979) [7] Luiiţa Leee Niulescu Usig he soluio of a absrac oe roble o solve soe classical cole oe robles Roaia Joural of Pure ad lied aheaics 5 (6) [8] L. Leee-Niulescu ad. Zlăescu Soe ew asecs of he L oe roble Revue Rouaie de ahéaiues Pures e liuées 55 3 () [9] J.. ihăilă O. Oleau ad C. Udrişe arov-e ad oeraor-valued ulidiesioal oe robles wih soe alicaios Roaia Joural of Pure ad lied aheaics 5 (7) [] L. Leee Niulescu Trucaed rigooeric ad Hausdorff oe robles for oeraors. I Oeraor Theor Suer Tiişara Jue 9-Jul 4 Thea Buchares () [] L. Leee Niulescu Posiive defiie oeraor-valued erels ad iegral rereseaios lied aheaics 3 () [] L. Leee Niulescu Sabili of oeraor-valued rucaed oe robles lied aheaics 4 4 (3) [3] O. Oleau New resuls o arov oe roble Hidawi Publishig Cororaio Ieraioal Joural of alsis Volue 3 ricle ID ages. h://d.doi.org/.55/3/938 [4] L. Gosse ad O. Ruborg Eisece uiueess ad a cosrucive soluio algorih for a class of fiie arov oe robles SI J. l. ah (8) [5] C. Berg ad. Thill Roaio ivaria oe robles ca aheaica 67 (99) [6] B. Fuglede The ulidiesioal oe roble. Eo ah I (983) [7] C. Kleiber ad J. Soaov ulivariae disribuios ad he oe roble Joural of ulivariae alsis 3 (3) [8] J. Soaov ad G. D. Li Hard s codiio i he oe roble for robabili disribuios. Theor Probab. l (3) (SI ediio) [9] R. Crisescu Ordered Vecor Saces ad Liear Oeraors Ed. cadeiei Buchares ad bacus Press Tubridge Wells Ke 976. [] C. Niculescu ad N. Poa Eleee de eoria saiilor Baach (Elees of Theor of Baach Saces) Ed. cadeiei Bucureşi 98. [] W. Rudi aliză reală şi coleá. Ediția a reia. (Real ad cole aalsis. Third Ediio) Thea Bucureşi 999. [] H. H. Schaefer Toological Vecor Saces Sriger 97.