O Multlers of Orlcz Saces حول مضاعفات فضاءات ا ورلكس Mahmud Masr Mathematcs Deartmet,. A-Najah Natoal Uversty, Nablus, Paleste Receved: (9/10/000), Acceted: (7/5/001) Abstract Let (, M, ) be a fte ostve measure sace, X a Baach sace, a modulus fucto ad f : X a strogly measurable fucto. he Orlcz sace s.he sace of Bocher -tegrable fuctos, L (, f : ( ) d ( 1 s L (, f : d(.also, L (, f : ess su. t Whe X s a Baach algebra we show that the multlers M ( L (, ) of L (, s L (, f ( a) ( ( a for all a 1 ad b 0.Also, M ( L (, ) = L (, f ( a ( a) ( for all a,b [0, ) whch geeralzes the secal case X beg the comlex umbers C.Whe (, M, ) s also o-atomc we show that f L (, for all f L (, ( ff lmsu. x ( ( Moreover, M( L (, ) = L (, ff lm su whe X s commutatve. x ( hs geeralzes the secal case = [0,1] ad X=C. f : (,M, ) ملخص ليكن فضاءا قياسيا موجبا ومنتهيا و X اقتران قياسي بقوة.فان فضاء ا ورلكس هو فضاء بناخ و اقتران مطلق القيمة و X. L (, f : ( f ( t ) ) d ( t )
O Multlers of Orlcz Saces = L (,.ا يضا L (, f : d( وفضاء بوخنر عندما 1 هو فضاء بناخ الجبري فسنثبت ان شاء اللة ا ن المضاعفات a 1 ) لكل a) ( ( عندما a و L (, X اذا كان. f : ess su t, L ( هي ل M ( L (, ) a ( لكل a و b في ( a) ( عندما L (, = M ( L (, ) كذلك.b 0 (,0 [ مما يعتبر تعميما للحالة الخاصة التي تكون فيها X هي الاعداد العقدية C. عندما يكون f لكل L (, فضاءا غير نووي ا يضا سنثبت ان شاء اللة ا ن (, M, ) اذا (, M ( L ( اذا وفقط ) = L (, وكذلك lm su اذا وفقط اذا f L (, x ( (. [0,1]= و X=C تبديليا وهذا يعتبر تعميما للحالة الخاصة X عندما يكون lm su x ( 1. Itroducto If s a strctly creasg cotuous subaddtve fucto o [ 0, ) ad satsfes ( 0 f x 0, the we call a modulus fucto. Let (, M, ) be a fte ostve measure sace,.e., s a set, M s a algebra ad s a ostve measure wth ( ). If X s a Baach sace, the a fucto s : X s called a smle fucto f ts rage cotas ftely may dstct ots x, x, 1 1..., x ad s ({ x }), 1,,..., are measurable sets. Such a fucto s ca be wrtte as s 1 x j, where, for j,, j 1,,...,. s the characterstc fucto of the set ad A fucto f : X s sad to be strogly measurable f there exsts a sequece { s } of smle fuctos such that lm s ( 0 a.e.
Mahmud Masr 3 he Orlcz sace L (, s the set of all strogly measurable fuctos f wth f ( ) d(. If for all f, g L (, we defe d( f, g) f g, the d s a metrc o L (, uder whch t becomes a comlete toologcal vector sace [1,.70]. For 1, L (, wll deote the Baach sace of (equvalece classes of) strogly measurable fuctos f such that d (. he orm L (, s gve by f 1 d (. he essetally bouded strogly measurable fuctos f form the Baach sace L (, wth orm gve by f ess su. t If s the modulus fucto ( x, 0 1, the L (, s the sace L ( (,. Sce [,.159], for ay modulus fucto, lm su (1), x x 1 t follows that L (, L (,. Whe X s a Baach algebra (see [5]) a multler of L (, s a strogly measurable fucto g: X such that gf L (, for all f L (,. We deote the set of all multlers of L (, by M ( L (, ). For X = C, the comlex umbers, M( L (, C) M( L ) were studed []. I ths aer we show that some of the results [] stll hold for M ( L (, ). A measurable
4 O Multlers of Orlcz Saces set A s called a atom f each of ts measurable subsets has measure ether 0 or (A). he measure sace (, M, ) s called o-atomc f t cotas o atoms. I [4,.1] t s show that f (, M, ) s a fte o-atomc measure sace ad 0< < ( ), the there exsts a measurable set such that ( ). Usg ths we show that ( lm su ff f L (, for all f L (, x ( whe (, M, ) s a fte ostve o-atomc measure sace. hs geeralzes the ma result [3] where =[0,1] ad X=C.. Multlers of L (, Lemma.1 Let X be Baach algebra.if g M ( L 1 (, ), the g L (,. Proof: Suose that g M ( L 1 (, ). Let f : X be gve by e for all t, where e s the ut elemet of X. he 1 f L (, ad Hece, g L (,. g ( d ( = g ( d ( = gf 1<. he ext results are geeralzatos of those []. Wthout loss of geeralty we ca assume that s a ubouded modulus fucto ad (1)=1. For f s bouded, the L (, s the strogly measurable fuctos. Also, f (1) 1, the we ca relace by. (1) heorem. If ( a) ( ( a for all a 1 adb 0, the
Mahmud Masr 5 M ( L (, ) = L (,, where X s a Baach algebra. Proof: Let g L (, ad f L (,.Choose a atural umber such that g. he sce X s a Baach algebra we have gf ( g ( ) d( ( ) d( f < ( g ( ) d( hus L (, M( L (, ). We ote that M ( L (, ) L (, sce f e s the ut elemet of X ad e for all t,the f L (, ad gf = g L (, for all g M ( L (, ). Next, for g M ( L (, ) ad f L 1 (, let g ~ ( ( g( ) e ad 1 h( ( ) for all t. he h d ( = f 1 <. hus, h L (, ad hece gh L (,. If A={t: g( >1}, the g~ f 1 ( g ( ) d( = A A ( g ( ) d( = ( g ( ) ( h( ) d( ( g ( ) ( h( ) d( + ( g( ) ( h( ) d( \ A ( g ( h( ) d( + ( h( ) d( gh + h <. \ A
6 O Multlers of Orlcz Saces hs shows that g ~ ( g ) e s a multler of L 1 (,. hus ~ g L (, X ) by lemma.1. hs mles that gl (, ad M( L (, ) L (,. herefore, M ( L (, ) = L (,. heorem.3 If ( a ( a) ( for all a, b [ 0, ), the M( L (, ) = L (, where X s a Baach algebra. Proof: As theorem.1 we have M ( L (, ) L (,. Let g L (,. he, for all f L (, gf ( g ( ) d( ( ( g ( ) ( )) d( = g + f < ( g ( ) d( herefore, g M ( L (, ).hus L (, = M ( L (, ). he followg s a geeralzato of the ma result [3] from the Lebesgue measure o [0,1]= ad the comlex umbers C to a oatomc, fte, ad ostve measure sace. heorem.4 Let (, M, ) be a fte ostve, o-atomc measure sace, ad be a modulus fucto. he ( lm su ff f L (, for all f L (, x ( where X s a Baach algebra. ( Proof: Suose lm su. he there exsts a creasg x ( sequece { x } such that x 1 1 ad ( x ) for all = 1,,3,. Sce (
Mahmud Masr 7 (, M, ) s a fte o-atomc measure sace by lemma 10.1 [4,.1] there exsts a sequece { } of arwse dsjot measurable sets such that Defe ( ) 1 ad ( ) for all =1,,3, ( 1 f : X by x ( t e for all t where e s the ut elemet of X. he ) f ( ) d( = ( x ( e ) d ( 1 1 = ( d ( 1 = ( ) ( ) ( ) x 1 = herefore, f L (,. Moreover, 1 f ( ) d( = ( ( x ( e) ) d( = ( d( 1 1 1 1 = ( ( ) ( ) ( x ) ( ) =. 1 1 hus, f L (,. herefore, f f L (, for all f L (,, the ( lm su. x ( Coversely, suose ad K such that ( lm su. he there exst costats M x (
8 O Multlers of Orlcz Saces ( x ) M ( for all x K. Let f L (, ad A={t: f( K}.he sce X s a Baach algebra f ( ) d( = ( A ) d( ( K ) ( ) + A ( + ( A ) d( ) d( M ( ) d( ( K ) ( ) + M f <. herefore, f L (, for all f L (,. Corollary.5 Let (, M, ) be a o-atomc, fte, ostve measure sace, X be a commutatve Baach algebra, ad be a modulus fucto. he M ( L (, ) = L (, ff ( lm su x ( Proof: Note that L (, s a algebra ff f L (, for all 1 f L (, follows from fg (( f g) f g ) ad the corollary follows from theorem.4.
Mahmud Masr 9 Refereces L, 0 1, J. 1] Deeb, W. ad Khall, R., Best aroxmato (, Arox. heory, 58, (1989), 68-77. ] Deeb, W., Multlers ad Isometres of Orlcz Saces, Proceedgs of the coferece o Mathematcal Aalyss ad ts Alcatos, Kuwat Uv. Kuwat, Feb. 18-1, (1985). 3] A.Hakawat, he Multler Algebra of Orlcz Saces, A-Najah Uv. J. Res., Vol. 1, (1998), 1-6. 4] Lght, W. ad Cheey, W., Aroxmato heory esor Product Saces, Lecture Notes Math. 1169, Srgler-verlag, Berl (1985). 5] W. Rud, Fuctoal Aalyss, McGraw-Hll, (1973).