DSpace at VNU: Multipliers for Generalized Entire Dirichlet Sequence Spaces

- 1999 VNU JOURNA OF SCIENCE (Mat Sci., t x v M U L T IP L IE R S e n t ir e FOR D IR IC H L E T G E N E R A L IZ E D s e q u e n c e sp a c e s T rin h D ao C h ie n Girl L hi Ed u c Ht i o i i i ìi ìd Tvri i i i i ng DepHi ' t I i i ent I I N T R O D U C T I O N Given a ipquencc (ÀA-) with A^ G c , < |Aa.| t +00 and p > , consider tho gpneraliz'i entire Dirichbt series oc ^CA.£'p{Ai.z), G C k=\ where coefficents (1.: an' fomplpx nunibers and Ep[z) is tho Mittag-Lefflei funct ion: ( r being the G a m m a function) , f r o r ( i + i) In [5] W( piovfd that if the spiios ( 1.1) convorges absolutely for all G c log In l in i s u p - ^ f — = - 00 a n d conversev, if t H coofficient.s of the s.n ios ( 1) satisfv coiiditioii ( 1.2 ) and if lo^A‘ |Aa- iiinsup — < -1-00 flien the serũs ( 1.1 couvvv^cs absolut(‘ly for all z e c N e x t , i n t h e C1 < + 00Ị -I I" = 0} Ti'inh o C h i e ĩ i Iu [5], 1)V us in g t h e saiiu' i n r t h o d as iu [2], \V(' Ị)rovO(l tliat Ả is a C()iu])l('t(' s('pariHlt)l(' noii-iiormablc nio ti iz a h lr space, whoK' th(' iiiPtiic is j;iv('u t),v ( ỉ I > ) = II" - ^’11.4 ; " = ("/ ) e I n t l i i s n o t e VV(' c o i i t i i n H ' t o s t u d y m u l t i p l i i ' i s t x ' t w i ' c n t h e s o s p a c e s a n d ot l i M s e q u i c n c i ' spaces oil spaces A ami c w v K'call th at for two S('qu(nic(’ spaces A' and y tlu' symbol (.V Y ) -li'iiotrs the s('(ỊU('ncp s p a c e o f m u lt i p li e r s from A' t o i (A' Y ) = P.R [1]) (a"A-) e 'i' foi all ( a ) € A'} It is obvious tliat if A'l c A\> and V'l c Yo- tlieii (X->- i ' l ) c (A'l, V'i) (1.4) Also it is clear t h a t , th e K o t h c du a l of a soqiiencp space is, in fact, th e soqtLK'iK'c spaco of mu ltipliei s from this space to /, i.e (-4, h) = A ' \ A qupstion arises: wliat aibout multipliois from A and c to I,, (0 < p < +oo) and vifc-veiHH'.' This is the sub ject i:)f the p m se n f note would like to expif'ss my deep g r a t i t u d e to Prof XRuyt-ii Van M a n a nd I)i L.f' Flai Khoi for helpful stiggostioiis ill th e p r e p a r a t i o n of thi s papol, II MULTIPLIERS FOR GENERALIZED ENTIRE rJIRICHLET S E Q U E N C E sp - \c?;s First \V(‘ not(' tli(' followiui> li'sult L e m m a 2.1 U e hrive A d i , c l o o C CẢ) < p < + 00 Wo provo the following l(>ninia.s L e m m a 2.2 W'e lìỉìvc a) C)cC l>)Cc A), c) c c (C„ C) Proof: a) Lf't (i/A.) e {A C) Sr.pposo th at (»A.) ệ c T h e n for a ib itra ry M > a n d foi a scquriicp (f,,), < e„ i 0, tlK'io exists an iiicrea.sing scquencf' (A-„) of positive such t h a t > M -£ n V/; > We defiue th e sequence (c^.) as follows nk Ck = 0, 1/2 if othorwise n - 1, , , M u ltọ lie r s f o r Generalizec E n tire lio n Wf* liaví' 1/2 < ;1I11sup(A/ , ay M = A/ n —oc -oc +00 S' ( a - ) G A How ev or we lavo liin sup |r*.u.|’/ 1-^*1'’ = limsup(|i/A > Ì1—»oo^ A »oo liinsup(A / - f -> oo n—oc / as M +OŨ This iiplies that (Cf, m.) ệ c w iid i leads to a rontradictioii T i p implications b) and c) aiP obvious □ N i w WP c a n p r o v e t h o f o l l c w i i i g r e s u l t T h e o e n i We have (^ = (/,„ C) = (/^ , C) = (C, C) = ( ^ , = ( ^ , Ì,,) = ( A /oo) = c PTOof.Vvom Loiiiina 2.1, Lenur.a 2.2 ami (1.4), it follows th at c {A I,) c ( A loc) c {A, C) c c, C’ c ( A and C’ c {C, c c T f tli(*oiPin is proved (/oc C) c (/;„ C) c ( A C) c c c we prove the followiiiJL e r n n i 2.3 We ỈIỈÌVC a)'/,, .4) c 4, hj C, U ) c A c)Ac i C, A) Proof: a)-irst, we Iiotp th at (q.) E a ÌỈ and only if {(ị) G A (with any appropriato choicp of the ow n ) Furthoi more, we can check th a t the soquencp (Aa-) satisfies Condition (1.3) if and aily if tlioir exists a > such that CXD < + 00 (2 1) *-=1 Nw let (;/*.) e {Ip, A) Suppose th a t (i^.) Ệ A , which means th a t {uị) ị A Then there eists M > such th a t for a sequence (£„) ị , there exists an increasing sequence (A:„) of)ositivo numbers such that T ỉ-inh D a o C h i i e n > M - £„ , V7Í Akn i > This implies th a t {£„ - A/)|Aa„ , Víí > Dofinp a sequoiicp (r^.) as follows: exp if k = k'n, ÌÌ — L 2, , Ck = ot herwiso, 0, where < i\I — a and a > is defiiKHl by (2.1) Then, we have Ck ^ e x p oc oc z_> i—J '( - i n ) aa-„ r] < _1 [(7 - < +00, Tl= i 77=1 due to (2.1) which shows th a t (ca-) € Ip However, lini sup log Aa k—*oc = lira sup log f'A-„ »A'„ p = lim s u p (7 - e„) = > - o o , n—*oo Aa-„ p which moans th at ịịct,-ÌIk-)’’) Ệ Ả or {ckiik-) ị Ả This is a contradiction Hf“iìC(’ {Ij,, ^ ) c b) Let (iu.) G (C, /oo)- Assume th a t (?/.*.) ị A , then t h n c exists au increasing s vq n fv m v (Ả'„) of positive' Iiuiiihois suc’li tliat lilll |»A.„ I " n —► OC' = + 00 ( ( Consider a soquonce (aO as follows: f A -„/|ỉ/a-J, Cị,- — < [ 0, if />■ A-;,, ÌÌ = 1, , , , othowiso Then we have = < +00 Ả—*oo A-—*oo duo to (2.2) and (1.3) Hence (ct) G c Howrvor sup | c A i/A-| A'> = sup l a - , , i/j.,, I = sup h„ ÍÍ > T7> Hence {Ckĩỉk) ị loc- a contradiction c) T he implication A c (C, A ) is obvious We can prove the following □ = 4-00 2.2 ) M'dtipliers f o r G e nera lized En tire T l e o r o m 2.2 W’v liiiv(> (C 1^ ) - {C I„) = (C (/^ ^ ) = P tio/: Fioiii l ci iim a 2.1 l.ciiiiiia 2,:i a n d ( 1, 1) it follows t h a t ^ c {C A) c {1^ A) c Til' ĩll('Ulf'ni is R f i n a i k A) c A □ riKHjH'in 2.1 a n d 2.2 for i h ( ‘ (Ji(linai \' Dil iclil('ĩ s('i i('s OÍ H' a n d sf'\'(Mal ('(Jiiiplrx varal)l('s wTvr ỊM‘0V(‘(1 in a n d [4ị R EFER EN C ES [1 J M Ai iil crs ou cV A L Shields C'cx'fficient I iiu lli pl ic is of B l o c h f u n c t i o n s , Tnnis Aiiier Math Soc 2 ( ) 255-205 Lc Ilai Khoi ilol oiiHHphic Diliclilot s e rie s iiis(>v(‘ial v a i i a h l c Math Scnriil 77(1995) 85-11)7 Lc Hai k h o i M u l t i p l i e r s for I^iiichlf't s c rie s in tli(' coiiiiili'x |)laii(' S o u t h - E a s t A s i a n Mat h Bull ( Ĩ(J aỊ)Ị)(‘a i ' ) ;4 \ j ' Hai l \h o i Coefficient m u l t i p l i ( 'i s f(ji SOUK' classc.s o f Diriclilct s c ri e s ill s('C(nal Í'()U1Ị)1(‘X \ariaỉ)l('s A c ỉ a Mdi ỉ i \'ỉ( t ì i af i i ỉ ( ‘ti ( í o apỊ)('ai ) l n n l D a o ( ’lu cn S c q u c u c c S])acc otCof'ffic'inith of o('iicializ(-(l (-ntiic Diiiclilct S('IÌ('S \ 'N Ư Journal o f S c i n u T X a t Sci., I X I V \ o K 9 ) 8-15 TAF CHI KHOA HỌC DHQGHN, KHTN, t x v , n‘’ l - 1999 M IA X r i ” CVA KHOXC; C;iAX 1)A\' i) iH ! ( 'iií j: r XCỈUYKX s r v íU).\c: l Y i n h Đ o C h ìố ii S (Ỉiỉía (lục viì Đ o ĨỈÌO (Ỉiỉì ĩ^ỉỉi V(/] l.ai klioiip, ^ian (là\- A \'à V, khoii^ gian ílã>' cua Iilian tir từ V \'ào y \ kv ( A ) ) (lìrtrc x;íc (lịnh ỉilur sau (A '.)') := {(///.): khuip ” ia:i (lãv A cái' liọ su c ù a cliuỗi Diriclilct s u v r ộ n g (lạiio Eậ ị G V(V‘A-) G A'} Xót Ỹ2 íroi io (ló hàin M i r t a o - Lofflor Q u a in o t ả klioiift nịaii A ' ' đ ố i n g ẫ u K ỏ t h e c ù a A t a t h ấ y ră:ií { A l / } ~ troiio (tó /i = {(///^.); |//^.| < oc} M ộ t c u hòi đ ặ t ra: kố t q u ả sõ n hư nao (lối vái khono »,ian (lãy n i a Iihán từ fừ A A ‘' vào rác khong «ian qupi t h u ọ ' khác, c h ẳ i i - han /,,(0 < p < đ ế n c c noi (luiiíi íló o o ) , / o o Iigirợc lại? Bài báo Iiày S(' đồ cập ... i ' c n t h e s o s p a c e s a n d ot l i M s e q u i c n c i ' spaces oil spaces A ami c w v K'call th at for two S('qu(nic(’ spaces A' and y tlu' symbol (.V Y ) -li'iiotrs the s('(ỊU('ncp... XRuyt-ii Van M a n a nd I)i L.f' Flai Khoi for helpful stiggostioiis ill th e p r e p a r a t i o n of thi s papol, II MULTIPLIERS FOR GENERALIZED ENTIRE rJIRICHLET S E Q U E N C E sp - c?;s... that CXD < + 00 (2 1) *-=1 Nw let (;/*.) e {Ip, A) Suppose th a t (i^.) Ệ A , which means th a t {uị) ị A Then there eists M > such th a t for a sequence (£„) ị , there exists an increasing sequence