ON THE SM -OPERATORS

Transcription

1 Soepara d O The SM-operators ON THE SM -OPERTORS Soepara Darawijaya Musli sori da Supaa 3 3 Matheatis Departeet FMIP UGM Yogyaarta eail: aspoo@yahoo.o Matheatis Departeet FMIP Uiversitas Lapug Jl. Soeatri Brodjoegoro No Badar Lapug eail: asoath@yahoo.o BSTRCT This is a partial part of our results i studyig geeralizatio of Hilbert-Shidt ad Carlea operators i Baah spaes. This proble a be doe if we preserve soe itrisi properties of Hilbert spaes ivolved; for exaples reflexivity ad separability. The result of the geeralizatio of Hilbert-Shidt operator will be alled SM-operator. Ifat alost all of properties of the SM-operator preserve alost all of propertie s of the Hilbert-Shidt operators. The appliatio o soe lassial Baah spaes will appear i the ext publiatios. Keywords: Orthooral Shauder basesseparable ad reflexive Baah spaes Hilbert- Shidt operator Maalah diteria 7 Septeb er 5. INTRODUCTION Oe of the ost iportat lasses of bouded operators ihe lass of Hilbert-Shidt operators. Let H ad H be Hilbert spaes. bouded operator : H H is alled a Hilbert-Shidt operator if there exists a orthooral bases { e } of H suh that e <. This defiitio ipliehat : H H is a Hilbert-Shidt operator if ad oly if : H H is a Hilbert-Shidt operator ; i this ase e d for every orthooral bases { e } of H ad { d } of H. Now questio arises whether suh a operator a be developed i Baah spaes. The aswer is positive wheever we preserve soe istrisi properties of the two Hilbert spaesi.e. reflexivity ad separability. The separability of Baah spae X io guaratee the existee of outable bases of a Baah spae X ad the reflexivity of a Baah spae X io guaratee that the bases of X is shriig (Zippi968). Further Johso.et al.(97) poited out that the existee of bases i the dual X does iply that also X has a bases see also (Dapa;Morrisso). More preisely if a separable ad reflexive Baah spae X has a shriig bases so doehe dual spae X. For exaple l p < p < has a bases (Shauder bases) but. PRELIMINRY l has ot. I what follows we shall always assue that the Baah spaes X Y ad Z are reflexive ad separable ored spae. Let X be the dual spae of ( X ) that ihe 49

2 Berala MIP 6() Jauari 6 olletio of all otiuously liear futioals o X. We always write xx to stad for x ( x) ad vie versa for every x X ad x X. sequee of liearly idepedet vetors { e } X is alled a Shauder bases of X if for every vetor x X there is uiquely sequee of salars { α } suh that x α e. Further for sipliity ad soe reaso we assue that e for every. We defie a sequee of vetor { e } X whih is alled biorthooral syste of { e } as follows: ( ) α xe e x e e α e e α for every N. It irue that e X for e is liear ad bouded e e for every ad e e for every. The sequee { e } fors a bases of the losed subspae [{ e }] X. Espeially we have [{ e }] X if ad oly if { e } is shriig i.e. li e x e o for every x X (Lidestrauss ad Tzafriri 996 Propositio.b.). gai i what follows we shall always assue that { e } ad { d } are orthooral Shauder bases or i short OSB of X ad Y respetively. If L ( X X) where L ( X Y) ihe olletio of otiuously liear operators fro Baah spae X ito Baah spae Y the operator L ( Y X ) is alled the adjoit operator of if for ay x X ad y Y have x y xy. The we have we e d where for every... < > e d e d e d d It iplies < e d > d. < e d > < e d >. The dual spae X also has a dual. It is usually deoted by X is alled the seod dual of ( X ) ad osists of all otiuously liear futioals o X. For ) eah fixed x X x f to be defie ( ) f ( x ) for all f i X. It is lear that x ) is a liear futioal o X ad sie ) x( f ) f ( x) f x we see that x ) i X. Hee we a defie a ap φ fro X ito X by lettig ( x) φ x ) for eah x i X. Sie for ay ozero eleet x i ( X ) there is a eleet f X suh that f ad f ( x ) x. Thus the ap is liear ad ( x) the φ x for eah x i X. s a osequee we have φ ( x) sup x φ( x) x sup xx x x. (.) Thus φ is also a isoetry ad sets up a ogruee betwee X ad X. The ored spae is ibedded X ito X by the aoial ibeddig φ i a isoetrially isoorfi way ad φ ( X ) X. Thus X a be osidered as the ored spae X. 5

3 Soepara d O The SM-operators 3. MIN RESULTS Based o the results of the last disussio we start with the followig defiitio. Defiitio. operator L ( XY ) is alled a SM-operator fro X ito Y if e d < for every OSB { } Y. d of e of X ad { } It is lear that if is a SM-operator the the uber : is oegative ad it does ot deped o the hoie of a OSB { e } of X ad a OSB { d } of Y. Let SM( XY ) be the olletio of SM-operators fro a Baah spae X ito a Baah spae Y. By Defiitio ad (.) we have the followig theore. Theore. operator L ( XY ) is a SM-operator if oly if is a SM-operator that is SM( XY ) if ad oly if SM( Y X ) ad e d for every OSB { e } of X ad { d } of Y. Theore 3. Let { e } ad { d } be a OSB of Baah spae X ad Y respetively. The ( i ) ( ) SM XY for every SM( XY ) ii ( ) is a Baah spae with ( ) respet to. iii If SM( XY ) the is opat. Proof: (i) For every x X we have ad x xe e x x e x e x < > d x x x d whih iplies. (ii). The spae SM( XY ) is a ored spae (ii.a). with respet to the or. for: for every SM( X Y). O (ull operator) (ii.b).for every salar α ad SM(XY) we have α ad α α α (ii.). For every B SM( XY ) we have ( ) + B + B e d e + Be d + Be d + Be d 5

4 Berala MIP 6() Jauari 6 or + Be d + B + B + B. The proof of the opleteess of the spae is as follows. Let { } SM( X Y) be a arbitrary Cauhy sequee. The for ay uber > there is a positive iteger suh that for every two positive itegers we have <. We wat to prove that there is SM( XY ) suh that li. Sie L ( X Y) is oplete ad there is L ( XY ) suh that < for every or li. Thus we have j < j e d d j e d j for ay itegers st ad. By lettig we have j e d d j j ( ) e d j for ay itegers st ad. Lettig s ad t we have j e d d j ( e ) d < j j for every. Therefore SM( X Y ) ad hee + ( ) i SM ( X Y ). Moreover < for every. Hee li. (iii) If SM( XY ) ad x X we have x x d d ad for every positive iteger we defie a operator B : X Y: Bx x d d. It is lear that B L ( XY ) B is a fiite ra operator ad li B. Therefore is a opat operator. Theore 4. Let XY ad Z be Baah spaes. If SM( XY ) ad B L ( Y Z) the B SM( X Z) ad B B. Proof: For every OSB { e } of X { d } Y ad { f j } Z we have Be f j B e f j j B B that is B SM( X Z) ad B B. Let SM( X ) ad L ( X ) stad for SM( X X) ad L ( X X) respetively. Cobiig the results of Theore 3 ad Theore 4 we have proved that SM(X) is algebra where is a ivolutio fro SM(X) ito SM(X) satisfyig : 5

5 Soepara d O The SM-operators ad ( ) ; ( ) B B ( α ) for every B SM ( X) α + B + B ad a real salar α as stated i the followig theore. Theore 5. Let X be a Baah spae havig a shriig OSB. The SM( X ) is a Baah algebra ad a ideal of L ( X ). CONCLUSION Geeralizatio of Hilbert-Shidt operators ito Baah spaes a be doe by preservig the istrisi properties of Hilbert spaes i.e. separable ad reflexivity. The results deoted by SM(XY) has i geeral the sae properties of those of Hilbert-Shidt operators. The biorthooral syste ad {({ e } { e }) : { } { } } e X e X {({ d } { d }) : { } { } } d Y d Y ihe ey to solve the oditio of orthoorality i Hilbert-Shidt operators used later i SM ( X Y ). For further wors we have bee usig the operator i lassial Baah spaes L ad l < p <. p p CKNOWLEDGEMENTS The seod author was partially supported by BPPS DIKTI. We also tha all the aoyous referees for readig the paper arefully. REFERENCES Coway J.B. 99. Course i Futioal alysis Spriger Verlag New Yor. Dapa P.S.. O strog M-bases i Baah spaes with PRI Collet. Math Johso W.B.RoshetalH.P. ad ZippiM. 97. O bases fiite-diesioal deopositios ad weaer strutures i Baah spaes Israel J. Math Lidestrauss J ad Tzafriri L 996. Classial Baah Spaes I ad II Spriger Verlag New Yor. Morriso T.J.. Futioal alysis: Itrodutio to Baah Spae Theory Joh Wiley & Sos. I. New Yor. Weida J. 98. Liear Operators i Hilbert Spaes Spriger Verlag. New Yor. Zippi M rear o bases ad reflexivity i Baah spaes Israel J. Math

6 Berala MIP 6() Jauari 6 54