Hypercyclic Functions for Backward and Bilateral Shift Operators. Faculty of Science, Ain Shams University, Cairo, Egypt 2 Department of Mathematics,

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1 Jourl of themtcs d Sttstcs 5 (3):78-82, 29 ISSN Scece ublctos Hyercyclc Fuctos for Bcwrd d Blterl Shft Oertors N Fred, 2 ZA Hss d 3 A orsy Dertmet of themtcs, Fculty of Scece, A Shms Uversty, Cro, Egyt 2 Dertmet of themtcs, Hgher Techologcl Isttute, th of Rmd Cty, Egyt 3 Dertmet of themtcs, Fculty of Egeerg, The Brtsh Uversty Egyt, El Sherou Cty, Egyt Abstrct: roblem sttemet: Gvg codtos for blterl forwrd d ulterl bcwrd shft oertors over the weghted sce of -summble forml seres to be hyercyclc Ths rovdes geerlto to the cse of Hlbert sce Aroch: We used hyercyclcty crtero d some relmry cocets for forml Luret seres d forml ower seres oreover we got beefts of some dulty roertes of bove metoed sces Results: We obted ecessry d suffcet codtos for blterl forwrd d ulterl bcwrd shft oertors to be hyercyclc Cocluso: The blterl forwrd shft oertor ws hyercyclc o the sce of ll forml Luret seres d the ulterl bcwrd shft oertor ws hyercyclc o the sce of ll forml ower seres uder cert codtos Key words: Blterl shft oertors, weghted shft oertors, hyercyclc fuctos, hyercyclc oertors INTRODUCTION A vector x Bch sce X s clled hyercyclc for bouded oertor T f the orbt T x : s dese X The frst exmles of { } hyercyclc oertors ered the sce of etre fuctos defed over the comlex le, edowed wth the comct-oe toology I 929 Brhoff [] essetlly showed the hyercyclcty of the trslto oertors T f () = f ( + ),, whle cle [2] roved the hyercyclcty of the dfferetto oertor The oto of hyercyclcty o Bch sces strted 969 wth Rolewc [3], who showed tht y sclr multle λ B of the ulterl bcwrd shft B s hyercyclc o l ( < ) d c, wheever λ > Kt hs thess wth ttle vrt closed sets for ler oertors, uversty of Toroto, determed codtos uder whch ler oertor s hyercyclc Ths result, commoly referred to s the hyercyclcty crtero, ws ever ublshed d few yers lter t ws redscovered broder form by Gether d Shro [5] Durg the lst yer's hyercyclcty crtero o Bch or Frechet sces hs ttrcted my mthemtcs worg ler fuctol lyss d very mortt cotrbutos to the toc hve bee mde [5-3] We use the hyercyclcty crtero of [] G Godefroy d J H Shro (99), to show tht the blterl forwrd shft oertors o the sce of ll forml Luret seres d the ulterl bcwrd shft oertors o the sce of ll forml ower seres re hyercyclc Our results the cse = 2 re comtble to tht gve by H Sls (995) o the sce l 2 (Z) d geerle those gve by [2] ATERIALS AND ETHODS Let { ()} be seuece of ostve umbers wth () = We cosder the sce of ll fuctos f () = such tht: () Corresodg Author: N Fred, Dertmet of themtcs, Fculty of Scece, A Shms Uversty, Cro, Egyt 78 f ( ) = < () f rges oly the oegtve tegers, these re forml ower seres, otherwse they re forml Luret

2 J th & Stt, 5 (3):78-82, 29 seres, where s orthoorml bss We shll () deote these sces by: H : ower seres cse : Luret seres cse Hece: The: * U e () = Defto: The oertor o defed by f () = f () s clled the blterl forwrd shft; furthermore, the verse of whch s the blterl bcwrd shft s the oertor B defed o f () by B f () = The rght verse of s the ulterl bcwrd shft oertor S defed o H by, f () f () Sf () = Notg tht the shft oertor defed over the + sce by R f () = s shft oertor, for ( + ) whch we get R = ( + ) Sce () >, Z, the by tg w = () () = w d () = ( w ) N d = = the oertor T defed by Te = w e + s jectve blterl forwrd weghted shft oertor o { } l ( ) x : x x e, x x Ζ Ζ Ζ = = = < wth weght seuece {w }, w = O the other hd the oertor V defed by Ve = e s jectve w bcwrd weghted shft oertor o seuece {w }, w = l ( Ζ ) wth weght roosto : The blterl forwrd shft oertor o s utrly euvlet to the jectve blterl forwrd weghted shft oertor T metoed bove Coversely, every jectve blterl forwrd weghted shft oertor T s utrly euvlet to ctg o roof: We defe the utry oertor U: l ( Ζ) by: Ue : = () Thus: 79 * U Ue = w e + So, s utrly euvlet to jectve blterl forwrd weghted shft oertor (wth the weght seuece { w }) Coversely, sce T s jectve blterl forwrd weghted shft wth weght w, the () for ll Wthout loss of geerlty [4], we my ssume tht the shfts hve ostve weghts ( ) Notg tht: We get: * TU = w e () * UTU = + roosto 2: The blterl bcwrd shft oertor B o s utrly euvlet to the jectve weghted shft ler oertor V metoed bove (wth the weght seuece {(w ) }) Coversely, every jectve weghted shft oertor s utrly euvlet to B ctg o E The roof of ths result s smlr to tht of roosto roosto 3: The owers of the oertor re ( + ) bouded o wth = su rovded tht () the suremum the rght hd sde exsts roof: For f d hece:, t s see tht f () = = ( + ) ( + ) f = su f = () () + ()

3 J th & Stt, 5 (3):78-82, 29 ( + ) su () O the other hd + () d ( + ) ( + ) so ; hece su d the () () result holds I the sme mer we c rove tht ( ) () B = su d S = su rovded tht () ( + ) the rght hd sde exsts ech cse roosto 4: A suffcet codto for the seres to be coverget s: () = lm su < < lm f () ( ( ) ) = ( ) (2) roof: Let f () = E Sce codto (2) s () suffcet for the covergece of the seres for () every, the by usg Hölder eulty we get: = f () () Ths s the reured result Oe c esly see tht the dul of { } () s E γ, where + = d γ = { γ ()} = ( ()) I fct, for y fuctol U o, there exst elemet g() such tht = b E γ γ() U(f ()) = f (),g() = b for y f () =, () where f (),g() deote the vlue of the fuctol g() o the elemet f () Accordg to ths E γ otto we get: RESULTS Now we gve ecessry d suffcet codtos for blterl forwrd shft oertor o the sce E s hyercyclc roosto 5: The blterl forwrd shft oertor o s hyercyclc f d oly f for every ε > d N, there exsts suffcetly lrge N, such tht for every : ( + ) () < ε d > () ( ) ε (3) roof: Assume tht s hyercyclc Sce the set of hyercyclc fuctos for s dese, the gve ε > d N, let < so tht / ( δ ) < ε d there s hyercyclc fucto f() for such tht: f () Usg the fuctols dul sce of such tht: () m γ(m) wth orm oe the f (), f () () γ() () We get:, > (4) > δ, (5) Sce f() s hyercyclc, we c choose >2 such tht: f () (6) () Let mles tht: be fxed Sce >2 eulty (6) m, () γ(m) = δ,m + ( + ) f (), = () γ ( + ) () 8

4 J th & Stt, 5 (3):78-82, 29 From (4) d (5) we get: ( + ) δ < < ε () δ O the other hd, eulty (6) mles tht: () f (), = () () ( ) γ Therefore: () δ δ > > > ( ) δ ε Coversely, let Y := s, Ζ be dese () subset of d let B : Y Y be the ler mg defed by: B : = () () (7) Notce tht B = B = Id Y It s esy to see tht ot-wsely o Y O the other hd by usg (3) we get: ( ) () () B = Thus the hyercyclcty crtero s stsfed d the roof s comlete DISCUSSION Results obted from roosto 5 c be led the ext roosto but for the ulterl bcwrd shft oertor o the sce H roosto 6: The ulterl bcwrd verse shft oertor S o H s hyercyclc f d oly f for every ε > d N, there exsts suffcetly lrge N, such tht for every N: roof: Assume tht S s hyercyclc Sce the set of hyercyclc fuctos for S s dese, the gve ε > d N, let < so tht / ( δ ) < ε d there s hyercyclc fucto f() for S such tht: f () = () Choosg sutble fuctols the dul sce of we get:, > (9) > δ, () Sce f () s hyercyclc, we c choose > such tht: S f () () () get: = Let be fxed From (9), () d () we Therefore: δ < + () () ( + ) ( + ) ( + ) δ < < ε () δ 2 Coversely let X := s,, be () (2) dese subset of H d let A : X X be the ler mg defed by: + A : = () () Notce tht ( ) (2) S A f () = f () d S otwsely o X O the other hd by usg (8) we get: ( + ) < ε () (8) 8 ( + ) = () = () A =

5 J th & Stt, 5 (3):78-82, 29 Thus the hyercyclcty crtero s stsfed d the roof s comlete CONCLUSION The blterl forwrd shft oertor s hyercyclc o the sce of ll forml Luret seres d the ulterl bcwrd shft oertor s hyercyclc o the sce of ll forml ower seres uder cert codtos roostos 5 d 6 the cse of = 2 re comtble wth tht gve [7] over the sce l 2 (Z) d [] over the sce l 2 (N) Notg tht, some exmles of hyercyclc bouded ler oertors hve lctos hyscs d utum rdto feld theory [,2] REFERENCES Brhoff, GD, 929 Demostrto due theoreme elemetre sur les fuctos eteres C R Acd Sc rs, 89: htt://wwwemsde/cgb/jf-tem? cle, GR, 952 Seueces of dervtves d orml fmles J Al th, 2: DOI: 7/BF Rolewc, S, 969 O orbts of elemets Stud th, 32: 7-22 jourlsmgovl/cgb/shvold?sm32 4 Dougls, RG, HS Shro d AL Shelds, 97 Cyclc vectors d vrt subsces for the bcwrd shft oertor A de l, Ist Fourer, 2: htt://wwwumdmorg/tem?d=aif_ _ 5 Gether, R d JH Shro, 987 Uversl vectors for oertors o sces of holomorhc fuctos roc Am th Soc, : htt://wwwjstororg/stble/ Bourdo, S, 993 Ivrt mfolds of hyercyclc vectors roc Am th Soc, 8: htt://wwwjstororg/stble/263 7 Sls, H, 995 Hyercyclc weghted shfts roc Am Soc, 347: htt://wwwjstororg/stble/ Asr, SI, 995 Hyercyclc d cyclc vectors J Fuct Al, 28: DOI: 6/JFAN Sls, HN, 999 Suercyclcty d weghted shfts Stud th, 35: htt://mtwbcmedul/s/sm/sm35/sm35 5df Ju Bes d Alfedo ers, 999 Heredtrly hyercyclc oertors J Fuct Al, 67: 94-2 DOI: 6/JFAN Rodrgue, A d HN Sls, 2 Suercyclc subsce: Sectrl theory d weghted shfts Adv th, 63: DOI: 6/AIA22 2 Emmrd, H d GS Heshmt, 25 Chotc weghted shfts Brgm sce J th Al A, : DOI: 6/JJAA Sls, H, 26 thologcl hyercyclc oertors Arch th, 86: DOI: 7/s3-5-5-y 82