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.