Cyclic behavior of linear fractional composition operators

E ex ac a ma hema icae Vol. 16, N´um. 1, 147 – 152 (2001) Cyclic Beha io o Linea F ac ional Composi ion Ope a o s E. A. Galla do-Gu i´ e ez, A. Mon es-Rod ´ ıguez Dp o. de Ma em´a icas, Uni . de C´adiz, Apa ado 40, 11510-Pue o Real, C´adiz, Spain Dp o. de An´alisis Ma em´a ico, Uni . de Se illa, A da. Reina Me cedes, Apa ado 1160, 41080-Se illa, Spain e-mail: e a.galla [email protected], amon [email protected] (Resea ch announcemen p esen ed by M. Gonz´alez) AMS Subjec Class. (2000): 47B38, 30D05 Recei ed Oc obe 20, 2000 1. Cyclic linea ac ional composi ion ope a o s Fo each eal numbe ν he weigh ed Di ichle space Sνis he space o analy ic unc ions (z) = P∞ n=0 anznsuch ha he ollowing no m k k2 ν= ∞ X n=0 |an|2(n+ 1)2ν is ini e. Obse e ha k · kνcomes om an inne p oduc and, he e o e, he spaces Sνa e Hilbe spaces. Also, no ice ha he abo e condi ion on he Taylo coe icien s o he unc ion implies ha lim supn→∞ |an|1/n ≤1 and, hus, each ∈ Sνis analy ic, a leas , on he uni disk Do he complex plane. Fo some alues o ν he spaces Sνa e e y well known classical analy ic unc ion spaces: o ν= 1/2 i is he Di ichle space D; o ν= 0 i is he Ha dy space H2and o ν=−1/2 i is he Be gman space A2. The easies composi ion ope a o s a e hose induced by linea ac ional maps ϕ(z) = az +b cz +dad −bc 6= 0 and ϕ(D)⊂D. In his case, he linea ac ional composi ion ope a o de ined by Cϕ = ◦ϕ( ∈ Sν) 147 148 e.a. galla do-gu i´ e ez, a. mon es- od ´ ıguez is always bounded on any o he Sνspaces (see [8], o ins ance). In some o he Sνspaces, his ac ails o be ue o ce ain holomo phic sel maps o D(see [4]). As weigh ed shi s, composi ion ope a o s a e a class o conc e e ope a o s ha , in gene al, do no all in o a class o ope a o s sa is ying a p esc ibed p ope y in a ian unde simila i y. This along se e al nice ea u es makes hem wo h s udying. A bounded linea ope a o Tac ing on a sepa able Hilbe space His said o be cyclic i he e is an ∈ H such ha he linea span o he o bi {Tn }n≥0is dense in H. The e ha e been much in e es o a long ime in cyclic ope a o s because hei ela ion o in a ian subspaces. Mo e ecen ly, s onge o ms o cyclici y a e being in es iga ed. I he e is ∈ H such ha {λTn :λ∈Cand n= 0,1, . . .}is dense in H, hen he ope a o Tas well as he ec o a e called supe cyclic. I he o bi i sel is dense in H, hen T and a e called hype cyclic. In [5] he au ho s ha e comple ely cha ac e ized he cyclici y and hype - cyclici y o scala mul iples o linea ac ional composi ion ope a o s in all he Sνspaces (see Table I and [2] o he classi ica ion o linea ac ional maps). P e ious esul s o ν= 0 and ν > 1/2 (o ν > 3/2, depending on which ype o ϕ) we e al eady known (see [3] and he page 150). Cyclic composi ion ope a o s o ν= 0 we e i s s udied in [13]. Obse e he cen al ole played by he Di ichle space Din he cu -o o any o he cyclic p ope ies. Since su- pe cyclici y is an in e media e p ope y be ween cyclici y and hype cyclici y, many o he esul s o he supe cyclici y column ollow immedia ely. I ϕhas a ixed poin in D(as in he ellip ic case), i is easy o show ha he e is no supe cyclic ope a o s (see [5, Sec . 8]). Fo he Di ichle space D(ν= 1/2), he supe cyclici y ollows because λCϕis hype cyclic whene e |λ|>1 (see [5, Sec . 4]). The e o e, o comple e he cha ac e iza ion i is jus su icien o know he supe cyclic beha io o composi ion ope a o s induced by pa abolic non au omo phisms. In his sense, Shapi o [11], using ha Cϕis simila o λCϕac ing on ce ain subspace, had p o ed ha λCϕis no hype cyclic on H2. Bu his is a o mean ha Cϕis supe cyclic on H2. We s ess he e ha e en an in e ible ope a o can be supe cyclic and no hype cyclic o any scala mul iple o i . This can be seen as a consequence o Theo em 5.2 in [7]. Be e s ill, Salas [10] has p o ided examples o his beha io in which e en he se o no mal eigen alues is he emp y se . The ollowing esul ha comple es he supe cyclic beha io o linea ac- ional composi ion ope a o s in he Ha dy space H2was i s ly p o ed in [6]. supe cyclic ope a o s 149 Theo em 1. Le ϕbe a pa abolic non au omo phism ha akes he uni disk in o i sel . Then Cϕac ing on he Ha dy space H2is no supe cyclic. Ske ch o he p oo . Pa abolic non au omo phisms ha ake he uni disk in o i sel ha e jus one ixed poin on he bounda y ∂Dand because any cyclic p ope y is in a ian unde simila i y i may be assumed ha ϕ(1) = 1 and, hus, he ollowing o mula holds ϕ(z) = (2 −a)z+a −az +2+a whe e <a > 0 because ϕis no an au omo phism o D. The me hod o he p oo is o ge es ima es om abo e and om below o Cϕac ing on ce ain subspaces. These subspaces a e buil up om he eigen unc ions o Cϕ. Fo each ≥0 an elemen a y compu a ion shows ha he inne unc ion e (z) = exp · z+ 1 z−1¸ is an eigen unc ion o Cϕin H2co esponding o he eigen alue e−a (see [4] o mo e abou he spec um o Cϕ). The p oo also uses he ollowing esul span {e : ≥0}=H2 ha is al eady con ained in Ahe n and Cla k’s wo k [1]. By aking quo ien s by he Cϕ-in a ian subspace F= span {eτ, eσ}, whe e τ > σ > 0, he space b H=H2/F can be decomposed in an o hogonal sum wi h some use ul p ope ies b H2=b Xτ⊕b Zτσ ⊕b Yσ.(1) Fi s , i σis chosen la ge enough, hen he e a e cons an s 0 < c < C such ha he ope a o b Cϕde ined by b Cϕˆ =d Cϕ sa is ies kb Cn ϕ¯¯ b Yσ ˆ k ≤ cnkˆ k o ˆ ∈b Yσand kb Cn ϕ¯¯ b Xτ ˆ k ≥ Cnkˆ k o ˆ ∈b Xτ. Second, he spaces in (1) a e b Cϕ-in a ian . The es ima es abo e can be ob ained by using Ge schgo in’s Theo em (see [12]) abou localiza ion o eigen alues. This heo em is applied o n×nma ices ha ep esen he es ic ions o b Cϕ o ce ain n-dimensional b Cϕ-in a ian subspaces. The ma ices a e ob ained hank o e y nice o hog- onali y p ope ies ha possess he eigen unc ions e (z) in he Ha dy space H2. 150 e.a. galla do-gu i´ e ez, a. mon es- od ´ ıguez Finally, suppose ha Cϕis supe cyclic, hen so is b Cϕ. Now, choose a supe cyclic ec o b =ˆ τ⊕ˆ τσ ⊕b σ. Then ˆ τmus be di e en om ze o; o he wise b Cϕis no supe cyclic. Finally, o ˆg6= 0 o hogonal o b Xτ, we ha e |h b Cn ϕˆ , ˆgi| kb Cn ϕˆ kkˆgk≤cnk σk Cnk τk ha goes o ze o. The e o e, Cϕis no supe cyclic, a con adic ion. The las idea in he ske ch o he p oo has occu ed i s in [9] in ela ion o in ini e dimensional subspaces o supe cyclic ec o s (see [6] o mo e de ails). Type o ϕCyclic Supe cyclic Hype cyclic Example Hype bolic Au omo phism ν < 1/2ν < 1/2ν < 1/23z+ 1 z+ 3 Pa abolic Au omo phism ν < 1/2ν < 1/2ν < 1/2(1 + i)z−1 z+i−1 Hype bolic Non- Au omo phism Always ν≤1/2ν < 1/21 + z 2 Pa abolic Non- Au omo phism ν≤3/2 Ne e Ne e 1 2−z In e io & Ex e io Always Ne e Ne e −z 2 + z In e io & Bounda y Ne e Ne e Ne e z 2−z Ellip ic I a ional Ro a ion Always Ne e Ne e e2i/3z Ellip ic Ra ional Ro a ion Ne e Ne e Ne e e2πi/3z Table 1 supe cyclic ope a o s 151 The compa ison p inciple o supe cyclic ope a o s ( i s no iced in [10]) only shows ha Cϕis no supe cyclic o any Sν o ν≥0. Howe e , a ca e ul analysis shows ha he me hods we ha e jus desc ibed can also be used o p o e ha Cϕis no supe cyclic in any o he Sνspaces. Al hough he e is no good o hogonali y p ope ies o he unc ions e (z) in Sν o ν > 0, one can s ill ob ain a he manageable Cϕ-in a ian subspaces. In ac , we can p o e he ollowing esul ha comple es he cyclic beha io o linea ac ional composi ion ope a o s in all he Sνspaces. Theo em 2. Le ϕbe a pa abolic non au omo phism ha akes he uni disk in o i sel . Then Cϕis no supe cyclic in any o he weigh ed Di ichle spaces Sν. Ac ually, ou me hods a e s ill alid o spaces o analy ic unc ions ha a e no Hilbe spaces, and can be used o ge non supe cyclici y o Cϕ, o ins ance, in he Be gman spaces Ap, 1 ≤p < ∞(and by he compa ison p inciple in any space ha is densely con ained in some o hese spaces and has a s onge opology). The compa ison p inciple is one o he mo i a ions o he wo k discussed in his no e (see [5] o mo e de ails). Re e ences [1] Ahe n, P.R., Cla k, D.N., On unc ions o hogonal o in a ian sub- spaces, Ac a Ma hema ica, 124 (1970), 191 – 204. [2] Ahl o s, L.V., “Complex Analysis”, McG aw-Hill, New Yo k, 1979. [3] Bou don, P.S., Shapi o, J.H., Cyclic composi ion ope a o s on H2, P oc. Symp. Pu e Ma h., 51 (2) (1990), 43 – 53. [4] Cowen, C., MacClue , B., “Composi ion Ope a o s on Spaces o Ana- ly ic Func ions”, CRC P ess, 1995. [5] Galla do-Gu i´ e ez, E.A., Mon es-Rod ´ ıguez, A., The ole o he spec um in he cyclic beha io o composi ion ope a o s, p ep in . [6] Galla do-Gu i´ e ez, E.A., Mon es-Rod ´ ıguez, A., The ole o he angle in he supe cyclic beha io , p ep in . [7] Gonz´ alez, M., Le´ on-Saa ed a, F., Mon es-Rod ´ ıguez, A., Semi-F edholm heo y: hype cyclic and supe cyclic subspaces, P oc. Lon- don Ma h. Soc., 81 (3) (2000), 169 – 189. [8] Hu s , P.R., Rela ing composi ion ope a o s on di e en weigh ed Ha dy spaces, A ch. Ma h., 68 (1997), 503 – 513. [9] Mon es-Rod ´ ıguez, A., Salas, H.N., Supe cyclic subspaces: spec al heo y and weigh ed shi s, p ep in . [10] Salas, H.N., Supe cyclici y and weigh ed shi s, S udia Ma h., 135 (1999), 55 – 74. [11] Shapi o, J.H., “Se ies o Lec u es in he Hype cyclici y/Subno mal Semi- na ”, Michigan S a e Uni e si y, Ap il, 1998. 152 e.a. galla do-gu i´ e ez, a. mon es- od ´ ıguez [12] Zalewska-Mi u a, A., Zem´ anek, J., The Ge schgo in discs unde uni a y simila i y o bi s, Linea ope a o s. Banach Cen e Publica ions, (1997), 427 – 441. [13] Zo boska, N., “Composi ion Ope a o s on Weigh ed Ha dy Spaces”, The- sis, Uni e si y o To on o, 1987. [14] Zo boska, N., Cyclic composi ion ope a o s on smoo h weigh ed Ha dy spaces, Rocky Moun ain Jou nal o Ma hema ics, 29 (1999) 725 – 740.