TITLE: HYPERCYCLIC SEQUENCES OF DIFFERENTIAL AND
ANTIDIFFERENTIAL OPERATORS.
AUTHOR: LUIS BERNAL-GONZ´
ALEZ.
AFFILIATION: DEPARTAMENTO DE AN´
ALISIS MATEM´
ATICO. FACUL-
TAD DE MATEM´
ATICAS. AVENIDA REINA MERCEDES. APARTADO
1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email p o ec ed].
FOOTNOTES TO THE TITLE: *This wo k is suppo ed in pa by DGICYT
g an PB96-1348.
1991 Ma hema ics Subjec Classi ica ion: P ima y 47B99. Seconda y 30E10,
32A07.
Key wo ds and ph ases: hype cyclic ope a o , hype cyclic sequence, F ´eche
space, in a ian linea mani old, analy ic unc ion o se e al complex a iables,
Runge domain, in ini e o de diffe en ial and an idiffe en ial ope a o s, ze o- ee
unc ion.
1
ABREVIATED TITLE: HYPERCYCLIC SEQUENCES.
NAME AND MAILING ADDRESS OF THE AUTHOR TO WHOM
PROOFS SHOULD BE SENT: LUIS BERNAL-GONZ´
ALEZ. DEPARTA-
MENTO DE AN´
ALISIS MATEM´
ATICO. FACULTAD DE MATEM´
ATICAS.
AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN.
E-MAIL: lb[email p o ec ed].
2
HYPERCYCLIC SEQUENCES OF DIFFERENTIAL
AND ANTIDIFFERENTIAL OPERATORS
By
LUIS BERNAL–GONZ´
ALEZ*
Abs ac . In his pape , we p o ide some ex ensions o ea lie
esul s abou hype cyclici y o some ope a o s on he F ´eche space
o en i e unc ions o se e al complex a iables. Speci ically, we gene-
alize in se e al di ec ions a heo em abou hype cyclici y o ce ain
in ini e o de linea diffe en ial ope a o s wi h cons an coefficien s and
s udy he co esponding p ope y o ce ain kinds o “an idiffe en ial”
ope a o s which a e in oduced in he pape . In addi ion, he exis ence
o hype cyclic unc ions o ce ain sequences o diffe en ial ope a o s
wi h addi ional p ope ies, o ins ance, boundedness o wi h some
non anishing de i a i es, is es ablished.
1. INTRODUCTION AND NOTATION
In his pape we deno e by N he se o posi i e in ege s, by C he ield o
complex numbe s and by N0 he se N0=N∪ {0}. Le X, Y be wo opological
spaces, Tn:X→Y(n∈N) a sequence o con inuous mappings and x∈X.
Then xis said o be hype cyclic (o uni e sal) o {Tn}i i s o bi {Tnx:n∈N}
unde {Tn}is dense in Y. The sequence {Tn}is hype cyclic whene e i has a
hype cyclic elemen . I is clea ha , in o de ha {Tn}can be hype cyclic, Ymus
3
be sepa able. I T:X→Xis a con inuous sel mapping, hen an elemen x∈Xis
said o be hype cyclic o Ti and only i i is hype cyclic o he sequence {Tn},
whe e Tn=T◦T◦... ◦T(n imes). Tis hype cyclic when he e is a hype cyclic
elemen o T. A subse A⊂Xis in a ian unde Twhen TA ⊂A. I is e iden
ha xis hype cyclic o Ti and only i he e is no p ope , closed, in a ian subse in
Xcon aining x. So, hype cyclici y is connec ed wi h he p oblem o he in a ian
subspace. I Xis a linea opological space, we say ha Tis an ope a o on X
whene e Tis a con inuous linea ans o ma ion aking Xin o i sel .
We now u nish a sufficien condi ion o a sequence {Tn} o be hype cyclic.
I s p oo is an easy applica ion o he Bai e Ca ego y Theo em and is le o he
eade . Se e al e sions o his esul ha e ea lie appea ed in [13, Sec ion 2], [14,
Sec ion 1], [15, Sa z 1.2.2], [16] and [19, Theo em 2.1]. No e ha , in a Bai e space
X, a dense Gδsubse is “ e y la ge” in X. A subse A⊂Xis esidual i and only
i i con ains a dense Gδsubse .
THEOREM 1. Le Xbe a linea opological space ha is a Bai e space, Ya
me izable sepa able (linea opological space, D⊂Xdense in X,E⊂Ydense
in Yand Tn:X→Y(n∈N)a coun able amily o con inuous linea mappings
sa is ying he ollowing condi ion:
Fo e e y d∈Dand e e y e∈E he e is a sequence {xk} ⊂ X
and a subsequence {nk}o posi i e in ege s such ha xk→dand
4
Tnk(xk)→e(k→ ∞).
Then {Tn}has a dense Gδsubse o hype cyclic ec o s.
The exis ence o hype cyclic ope a o s on any sepa able F ´eche space has been
ecen ly p o ed in [1] (see also [7]). B. Beauzamy [2, 3, 4] has cons uc ed examples
o linea ope a o s on Hilbe spaces ha ing dense, in a ian linea mani olds all o
whose nonze o elemen s a e hype cyclic. P. S. Bou don [11] p o ed in 1993 ha
any hype cyclic ope a o on a complex Banach space has a dense, in a ian linea
mani old consis ing, excep o ze o, en i ely o hype cyclic ec o s. In ac (see [1])
his esul holds in a mo e gene al se ing. We s a e i o u u e e e ences.
THEOREM 2. Le Tbe a hype cyclic ope a o on a complex, sepa able, locally
con ex space X. Then he e is a dense T-in a ian linea mani old o Xconsis ing
en i ely, excep o ze o, o ec o s ha a e hype cyclic o T.
Le Gbe a nonemp y open subse o CN(N∈N). Gis said o be a domain
when, in addi ion, i is connec ed. A domain G⊂CNis said o be a Runge domain
i each analy ic unc ion on Gcan be app oxima ed uni o mly by polynomials on
e e y compac subse o G(see [18, pp. 52-59] and [20, Chap e 5]). When N= 1,
he Runge domains a e p ecisely he simply connec ed domains. Deno e by H(G),
as usual, he F ´eche space o analy ic unc ions on Gendowed wi h he compac -
open opology. G. D. Bi khoff [10] showed in 1929 ha e e y ansla ion ope a o
5

τa( ha is, τa (z) = (z+a), whe e a∈C {0}is ixed) is hype cyclic on he
space H(C) and G. R. MacLane [23] ob ained he same conclusion in 1952 o he
ope a o o diffe en ia ion 7→ ′. G. Gode oy and J. H. Shapi o [14, Sec ion 5]
demons a ed in 1991 he ollowing s ong gene aliza ion o he heo ems o Bi khoff
and MacLane:
THEOREM 3. I Lis an ope a o on he space H(CN)o en i e uncions on
CN ha commu es wi h each o he ansla ion ope a o s τa(a∈CN), and is no a
scala mul iple o he iden i y, hen Lhas a dense, in a ian ec o mani old each
o whose non-ze o elemen s is hype cyclic o L.
See also [5, 8, 9, 15 and 22] o o he gene aliza ions o Bi khoff-MacLane’s
heo ems. Se e al wo ks ha e been made in connec ion wi h addi ional p op-
e ies imposed o hype cyclici y. Fo ins ance, G osse-E dmann [16] p o ed in
1990 ha he e is no hype cyclic en i e unc ion o he diffe en ia ion ope -
a o Dsa is ying max|z|= | (z)|=O(e / 1/2) ( → ∞), while he e is a D-
hype cyclic en i e unc ion such ha max|z|= | (z)|=O(φ( )·e / 1/2) ( → ∞),
φ: (0,+∞)→(0,+∞) being a p e ixed unc ion such ha φ( )→ ∞ ( → ∞).
G. He zog [17] showed in 1994 ha he e is a D-hype cyclic unc ion such ha
and ′a e ze o- ee. This esul has been ecen ly imp o ed by he au ho [6], which
p o es ha , i q∈N0and a noncons an en i e unc ion Φ o subexponen ial ype
6
a e gi en, hen he se A={ ∈H(C) : (q)and (q+1) a e ze o- ee}con ains
a esidual subse o Φ(D)-uni e sal unc ions. The esul is sha p in e ms o he
g ow h and he ype o Φ.
In his pape we ex end Theo em 3 and he esul o he la e pa ag aph
abou ze o- ee de i a i es o mo e gene al domains and sequences o ope a o s
and in oduce and s udy a new kind o ope a o s ela ed o an ide i a i es. The
exis ence o bounded hype cyclic unc ions is es ablished o ce ain domains. We
also p o ide a a he gene al “eigen alue es ” in o de o p o e he hype cyclici y
o ce ain kinds o ope a o s and sequences o ope a o s.
2. DIFFERENTIAL AND ANTIDIFFERENTIAL OPERATORS
In o de o gene alize in Sec ion 4 Gode oy-Shapi o’s esul s a ed in Sec ion
1, we adop he no a ion o [14, Sec ion 5] and ansc ibe some p elimina ies om
i . Fo 1 ≤j≤Nle Djdeno e complex pa ial diffe en ia ion wi h espec o he
j h coo dina e. A mul i-index is an N- uple p= (p1, ..., pN) o nonnega i e in ege s.
Deno e |p|=p1+... +pN,p! = p1!·... ·pN!, Dp=Dp1
1◦... ◦DpN
N(D0=I= he
iden i y ope a o ) and zp=zp1
1·... ·zpN
Ni z= (z1, ..., zN). An en i e unc ion
Φ(z) = ∑|p|≥0apzpon CNis said o be o exponen ial ype whene e he e exis
posi i e cons an s Aand Bsuch ha |Φ(z)| ≤ AeB|z|(z∈CN). This happens i
7
and only i he e is R∈(0,+∞) o which
|ap| ≤ R|p|
p!(|p| ≥ 0).
I is shown in [14] ha , i Φ is o exponen ial ype, hen he mapping Φ(D) =
∑|p|≥0apDpis a well-de ined ope a o on H(CN). No e ha i Φ is an en i e
unc ion and L= Φ(D), hen Ln= Φn(D) o all n∈N(Ln= L ◦L◦. . . ◦Lbu
Φn= Φ ·Φ·. . . ·Φ, n imes).
T i ially, e e y linea diffe en ial ope a o wi h cons an coefficien s commu es
wi h ansla ions. In [14] i is shown ha he ope a o s on H(CN) commu ing wi h
ansla ions beha e as “in ini e o de ” diffe en ial ope a o s.
THEOREM 4. Le Lbe an ope a o on H(CN). The ollowing condi ions a e
equi alen :
a) Lcommu es wi h e e y ansla ion ope a o τa(a∈CN).
b) Lcommu es wi h each o he diffe en ia ion ope a o s Dk(1 ≤k≤N).
c) L= Φ(D), whe e Φis an en i e unc ion on CNo exponen ial ype.
Some addi ional no a ions and esul s a e needed in o de o p o e ou heo-
ems. I a= (a1, ..., aN)∈CNand > 0, we deno e by D(a, ) he closed polydisc
D(a, ) = {z∈CN:|zj−aj| ≤ , 1≤j≤N}. We conside in CN he dis-
ance d(z, a) = max{|z1−a1|, ..., |zN−aN|}. I gis a unc ion de ined on a subse
B⊂CN, hen ||g||Bwill s and o sup{|g(z)|:z∈B}. We say ha an en i e unc-
8
ion Φ(z) = ∑|p|≥0apzpon CNis o subexponen ial ype whene e , gi en ε > 0,
he e exis s a posi i e cons an K=K(ε) such ha |Φ(z)| ≤ Keε|z|(z∈CN).
A s aigh o wa d compu a ion wi h powe se ies and he Cauchy inequali ies [18,
p. 27] shows ha Φ is o subexponen ial ype i and only i , gi en ε > 0, he e is a
posi i e cons an A=A(ε) such ha
|ap| ≤ A·ε|p|
p!(|p| ≥ 0).
No e ha , i N= 1, hen Φ is o subexponen ial ype i and only i Φ is ei he
o g ow h o de less han one o o g ow h o de one and g ow h ype ze o. Each
en i e unc ion o subexponen ial ype is ob iously o exponen ial ype.
THEOREM 5. I G⊂CNis a nonemp y open subse and Φ(z) = ∑|p|≥0apzp
is an en i e unc ion o subexponen ial ype, hen he se ies Φ(D) = ∑|p|≥0apDp
de ines an ope a o on H(G).
P oo . I G=CN, he esul is a pa icula case o he abo e conside a ions.
So, we may suppose ha G=CN. Fix ∈H(G) and a compac subse K⊂G.
Le ε=1
2d(K, CN G). Then he e is A∈(0,+∞) such ha |ap| ≤ A·(ε/2)|p|
p!
(|p| ≥ 0). Fix a poin a∈K. The Cauchy o mula o de i a i es [18, p. 27, Fo mula
2.2.3] ells ha
|Dp (a)| ≤ p!|| ||D(a,ε)
ε|p|≤p!|| ||K1
ε|p|,
whe e K1is he compac se {z:d(z, K)≤ε}. No e ha K⊂K1⊂G. The e o e
9
such ha lim
k→∞ Φnk(a) = 0 o all a∈F1and lim
k→∞ Φnk(b) = ∞ o all
b∈F2.
(Q) m(Φn)→ ∞ (n→ ∞) and he e is a nonemp y open subse B⊂
CNsuch ha o e e y ini e subse F⊂B he e exis s a subsequence
{nk}o posi i e in ege s sa is ying limk→∞ Φnk(b) = ∞ o all b∈F.
(Q’) m(Φn)→ ∞ (n→ ∞) and he e is a subse B⊂CNwi h a
leas one ini e accumula ion poin such ha o e e y ini e subse
F⊂B he e exis s a subsequence {nk}o posi i e in ege s sa is ying
limk→∞ Φnk(b) = ∞ o all b∈F.
T i ially (P) implies (P’) and (Q) implies (Q’). Fo ins ance, he sequence Φn(z) =
zn(z∈C;n∈N) sa is ies all ou p ope ies; he sequence Φn(z) = nnznsa is ies
(Q) bu does no (P) (nnzn→ ∞ as n→ ∞ o e e y z∈C {0}); he sequence
Φn(z) = nenz +zn
n2sa is ies (P) ( ake A={z:|z|<1,Re z < 0}and B={z:
|z|<1,Re z > 0}) bu no (Q).
THEOREM 8. Suppose ha Gis a Runge domain o CNand Φ,Φn(n∈N)
a e en i e unc ions on CN. Assume ha Φis no a cons an and deno e Ln=
Φn(D) (n∈N).
a) Suppose ha e e y Φnis o subexponen ial (exponen ial, esp.) ype and he se-
quence {Φn}sa is ies (P). Then he e is a dense Gδsubse o H(G)(H(CN), esp.)
16

all o whose elemen s a e hype cyclic unc ions o {Ln}.
b) Fo N= 1 he s a emen o a) s ill holds i (P) is changed o (P’).
c) Suppose ha Φis o subexponen ial ype and le L= Φ(D). Then he e is a
dense Gδsubse Mo H(G)all o whose elemen s a e hype cyclic unc ions o
L. In addi ion, Mcon ains all nonze o unc ions o a dense, L-in a ian , linea
submani old o H(G).
d) Suppose ha Lis an ope a o on H(CN) ha commu es wi h each o he ans-
la ion ope a o s τa(a∈CN), and is no a scala mul iple o he iden i y. Then
Lhas a dense Gδsubse Mo hype cyclic unc ions. In addi ion, Mcon ains all
nonze o unc ions o a dense, L-in a ian , linea submani old o H(CN).
P oo . a) Fi s ly, by Theo em 5 and he ini ial conside a ions o Sec ion 2,
e e y Lnis an ope a o de ined on H(G) (e en on H(CN) i Φnis o exponen ial
ype). F om now on, Gmay be CNo no . No e ha Djea=ajea o each
j∈ {1, ..., N}and each a∈CN, so Dpea=apea o e e y mul i-index p. Then
Lnea= Φn(D)ea= Φn(a)ea(a∈CN, n ∈N). Obse e ha each unc ion eais an
eigen ec o o e e y Lnwi h eigen alue Φn(a).
Conside he open subse s Aand Bp o ided by he condi ion (P). Fix a com-
pac subse K⊂G, a uncion ∈H(G) and ε > 0. Since Gis a Runge do-
main, a polynomial P(z) o Ncomplex a iables can be ound in such a way ha
| (z)−P(z)|< ε/2 o all z∈K. The e exis s h∈HS(S=Ao B) wi h
17
|P(z)−h(z)|< ε/2 o all z∈K. The e o e | (z)−h(z)|< ε o all z∈K. This
shows ha HAand HBa e also dense subse s o H(G). I now suffices o apply
pa 1) o Theo em 7 on X=H(G), A={ea:a∈A},B={eb:b∈B}and
Tn=Ln(n∈N).
b) This pa is ob ious om a), oge he wi h he ema k o he case N= 1
a he beginning o his sec ion. We would ha e anew ha HAand HBa e dense
in H(C), so in H(G) as well.
c) By Theo em 5, Lis an ope a o de ined on H(G). Since Φ is a noncons an
en i e unc ion, he se s A= Φ−1(|z|<1) and B= Φ−1(|z|>1) a e nonemp y
open subse s. Now use pa 2) o Theo em 7 wi h X=H(G), T=L= Φ(D),
A={ea:a∈A}and B={eb:b∈B}. No e ha , like in pa a), e e y unc ion
ea(a∈CN) is an eigen ec o o Twi h eigen alue λ(T, ea) = Φ(a).
d) This is essen ially Theo em 3. I has been pu he e o he sake o comple e-
ness. I is de i ed as c) (G=CNhe e) by using Theo em 4. I should be no ed
ha , i L= Φ(D), hen Φ is noncons an i and only i Lis no a scala mul iple o
he iden i y. ////
Fo ins ance, we ha e ha he e is a dense Gδsubse o en i e unc ions on
Csuch ha each en i e unc ion can be uni o mly app oxima ed on compac se s
by unc ions o he o m n (z+n) + (n)(z)
n2(n∈N). Indeed, i suffices o conside
18
he sequence Φn(z) o he hi d example jus be o e he la e heo em. No e ha
eaD =τa o e e y a∈C.
In iew o he esul on g ow h o G osse-E dmann [16] o en i e unc ions
(see Sec ion 1), i is na u al o ask wha is he minimal g ow h allowed o a D-
hype cyclic unc ion on a bounded domain in C. The answe o Runge domains
is almos i ial and is p o ided in Co olla y 2. We deno e, as usual, by g|S he
es ic ion o a unc ion g o a subse S.
COROLLARY 1. Assume ha G⊂CNis a Runge domain and ha Lis
an ope a o on H(CN) ha commu es wi h each o he ansla ion ope a o s τa
(a∈CN), and is no a scala mul iple o he iden i y. Then he se
M={ |G: is en i e and {(Ln )|G}∞
1is dense in H(G)}
is dense in H(G).
P oo . The asse ion is e iden om pa d) o Theo em 8 and om he ac
ha H(CN) is dense in H(G). ////
COROLLARY 2. I G⊂CNis a bounded Runge domain, hen he e exis s a
dense subse Min H(G)such ha , o e e y ∈M, each de i a i e (n)(n∈N0)
is bounded and he o bi { (n)}∞
1is dense in H(G).
P oo . Jus apply Co olla y 1 wi h L=D. ////
19
THEOREM 9. Suppose ha Gis a Runge domain o CNand Φ,Φn(n∈N)
a e en i e unc ions on CN. Deno e Ln= Φn(D) (n∈N).
a) Suppose ha e e y Φnis o subexponen ial (exponen ial, esp.) ype and he
sequence {Φn}sa is ies (Q). Then he e is a dense Gδsubse o H(G)(H(CN),
esp.) all o whose elemen s a e hype cyclic unc ions o {Ln}.
b) Fo N= 1 he s a emen o a) s ill holds i (Q) is changed o (Q’).
P oo . We can also apply pa 1) o Theo em 7. Take X=Y=H(G),
Tn=Ln(n∈N), A={zp:p∈N0N},B={eb:b∈B}whe e he se
Bis u nished by hypo hesis (Q) (o by (Q’) i N= 1). Obse e ha span A
(= {polynomials}) is dense in H(G). Each unc ion ebis an eigen ec o o e e y
Tnwi h eigen alue λ(Tn, b) = Φn(b). Fix wo ini e subse s F1={zp1, ..., zp } ⊂ A
and F2={eb1, ..., ebs}⊂B. F om (Q) (o (Q’)), a subsequence {nk}o posi i e
in ege s can be ound o he ini e se F={b1, ..., bs} ⊂ Bin such a way ha
limk→∞ λ(Tnk, bj) = limk→∞ Φnk(bj) = ∞ o all j∈ {1, ..., s}. On he o he hand,
i α= max{|p1|, ..., |p |}, he e is n0∈Nsuch ha m(Φn)> α o all n > n0, so
Φn(D)zpj= 0 o all j∈ {1, ..., }because Dpzpj= 0 o all j∈ {1, ..., }and o
e e y mul i-index pwi h |p|> α. Consequen ly, each zpjis an eigen ec o o Tnk
(we can assume nk> n0 o all k) wi h eigen alue λ(Tnk, zpj) = 0, which i ially
ends o ze o as k→ ∞. ////
20
Un o una ely, one canno expec any hype cyclici y esul o an an idiffe en-
ial ope a o Ψ(D−1).
THEOREM 10. Assume ha G⊂Cis a simply connec ed domain. Fix a poin
a∈Gand conside he co esponding an ide i a i e ope a o D−1. Suppose ha Ψ
and Ψn(n∈N)a e in S(1/∆(a, G)) and ha Ψn(z) = ∑∞
j=0 c(n)
jzj. Le L, Lnbe
he ope a o s L= Ψ (D−1), Ln= Ψn(D−1) (n∈N). We ha e:
a) I {Ln}is hype cyclic, hen he sequence {c(n)
0:n∈N}is dense in C.
b) Lis no hype cyclic.
P oo . Land Ln(n∈N) a e well de ined ope a o s by Theo em 6. I ∈H(G)
is hype cyclic o {Ln} hen, gi en b∈C, some subsequence {Lnk }o {Ln }
mus app oxima e he cons an unc ion g(z)≡bon he compac se {a}. Bu
(Ln )(a)=(∑∞
j=0 c(n)
jD−j )(a) = c(n)
0 (a) because D−j (a) = 0 o all j∈N.
This implies ha c(nk)
0 (a)→b(k→ ∞), so {c(n)
0:n∈N}is dense in C.
This p o es a). Pa b) is an unpleasan consequence o a): indeed, assume ha
Ψ(z) = ∑∞
j=0 cjzjand pu Ln=Ln. Then c(n)
0=cn
0and o each c0∈C he
sequence {cn
0}is no dense in C. ////
Ne e heless, a so o “pseudo-hype cyclici y” is ue, as ou nex heo em
shows. Fo his, le us ci e he ollowing esul o W. Luh [21]: Fo e e y simply
connec ed domain G⊂C he e exis s a sequence {Cn}∞
1⊂Cwi h he p ope y
21

ha o e e y φ∈H(G) he se {Qn(z) = D−nφ(z) + ∑n−1
j=0
Cn−j
j!zj:n∈N}is
dense in H(G). No e ha he coefficien s Cn’s do no depend upon φ.
Jus a ema k be o e he heo em. Le G⊂Cbe a simply connec ed domain
and ix a poin a∈G. I Ψ(z) = ∑∞
j=0 cjzjis a o mal powe se ies, hen Ψ ∈
S(1/∆(a, G)) i and only i α(Ψ, δ) is ini e o all δ∈(0,∆(a, G)), whe e we ha e
se
α(Ψ, δ) = |c0|+ sup
j∈N
|cj|δj−1
(j−1)! .
THEOREM 11. Assume ha G⊂Cis a simply connec ed domain. Fix a
poin a∈Gand conside he co esponding an ide i a i e ope a o D−1. Then
he e exis s a sequence {Cn}∞
1⊂Csa is ying he ollowing p ope y: Fo e e y
∈H(G)and e e y sequence {Ψn(z)}∞
1⊂S(1/∆(a, G)) o o mal powe se ies o
which
α(Ψn, δ)→0 (n→ ∞) o all δ∈(0,∆(a, G)),
he sequence {Ψn(D−1) (z) + ∑n−1
j=0
Cn−j
j!zj:n∈N}is dense in H(G).
P oo . Deno e Ln= Ψn(D−1) and assume ha Ψn(z) = ∑∞
j=0 c(n)
jzj(n∈N).
F om Theo em 6 we ha e ha e e y Lnis an ope a o on H(G). We apply he
men ioned esul o Luh [21] o he domain Gand he unc ion φ= 0. We ob ain
ha he e is a sequence {Cn}∞
1⊂Csuch ha he se {Hn}∞
1gi en by Hn(z) =
∑n−1
j=0
Cn−j
j!zjis dense in H(G). Fix a unc ion ∈H(G) and a compac se K⊂G.
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I he inal s eps o he p oo o Theo em 6 a e wa ched hen one sees ha he e
a e a compac se L⊂Gand posi i e cons an s σ, M, M1wi h M1∈(M, ∆(a, G))
such ha ||Ln ||K≤Bn· || ||L, whe e
Bn=|c(n)
0|+σM1
M1−M·sup
j∈N
|c(n)
j|Mj−1
1
(j−1)! o all n∈N.
Bu α(Ψn, M1)→0 (n→ ∞) by hypo hesis, so limn→∞ Bn= 0 and ||Ln ||K→0
(n→ ∞). Hence {Ln }∞
1con e ges uni o mly o ze o on compac se s. Since
{Hn}∞
1is dense in H(G), we ha e ha {Ln +Hn}∞
1is also dense in H(G), as
equi ed. ////
By using [21, Lemma 3] one can easily es ablish ha o e e y compac se B⊂
Cwi h connec ed complemen and e e y unc ion gwhich is con inuous on Band
holomo phic in he in e io o B, he e is a subsequence o {Ln +Hn}∞
1con e ging
o guni o mly on Band, in addi ion, o e e y Lebesgue-measu able se E⊂G
and e e y Lebesgue measu able unc ion g:E→C∪ {∞}, he e is a subsequence
o {Ln +Hn}∞
1con e ging almos e e ywhe e o gon E. Theo em 11 oge he
wi h his ema k gene alizes [5, Theo em 5]: in ac he e we deal wi h he case
Ψn(z) = cnzn, whe e {cn}∞
1⊂Cis a sequence such ha lim supn→∞(|cn|
n!)1/n ≤
1/∆(a, G).
We p opose he e as an open p oblem o gi e condi ions on {Ψn}which gua -
an ee he hype cyclici y o {Ψn(D−1)}. No e ha his sequence can ce ainly be
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hype cyclic. Indeed, o a= 0 he cons an unc ion ≡1 is {Ψn(D−1)}-uni e sal
i and only i he se {∑∞
j=0(c(n)
j/j!)zj:n∈N}is dense in H(G). App op ia e
c(n)
jcan always be ound.
To inish, we es ablish a esul abou hype cyclici y o unc ions wi h he addi-
ional p ope y ha ce ain de i a i es do no anish on he domain. No ice ha
he e is no D-hype cyclic en i e unc ion such ha · ′· ′′ is ze o- ee, since
{ ∈H(C) : · ′· ′′ is ze o- ee}={eαz+β:α, β ∈C, α = 0}(see [12, p. 433]
and [24]). I q∈N0, le us deno e A(q) = { ∈H(G) : (q)(z) (q+1)(z)= 0 o all
z∈G}. Since exp ∈∩q∈N0A(q), e e y A(q) is nonemp y. I L={Ln:n∈N}is
a sequence o con inuous mappings om Xin o Y, hen we deno e by HC(L) he
se o hype cyclic elemen s o L. I A⊂X, deno e L|A={Ln|A:n∈N}. We a e
now eady o s a e ou heo em on hype cyclici y. He zog’s esul [17] is he special
case q= 0, G=C,L={Dn:n∈N}while he esul o he au ho in [6] is he
special case G=C,L={Ln:n∈N}wi h L= Φ(D).
THEOREM 12. Assume ha G⊂Cis a simply connec ed domain and ha
{Φn:n∈N}is a sequence o en i e unc ions o subexponen ial ype sa is ying (P’)
o (Q’). Fix q∈N0and se A=A(q),L={Ln:n∈N}, whe e Lnis he ope a o
on H(G)gi en by Ln= Φn(D) (n∈N). Then he se HC(L|A)is esidual in A.
P oo . The p oo is e y simila o ha in [6], so we me ely indica e some
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necessa y changes. The closed discs D(0, k) (k∈N) in he p oo o Theo em
5 o [6] should be eplaced o he closu e Gko Gk, whe e {Gk:k∈N}is a
sequence o simply connec ed domains such ha e e y Gkis compac , Gk⊂Gk+1
and G=∪∞
k=1 Gk. The me ic d( , g) in ha pape is he e changed o
d( , g) =
∞
∑
j=1
1
2j
|| −g||j
1 + || −g||j
( , g ∈H(G)),
whe e ||h||j= maxz∈Gj|h(z)|. F om Theo ems 8 and 9, HC(L) is esidual in H(G)
and an adequa e applica ion o Theo em 2.1 o [17] will gi e he esul i one akes
in o accoun ha A=∩∞
k=1 Akwhe e Ak={ ∈H(G) : minGk| (q)· (q+1)|>0}.
I k∈Nis ixed, hen he e is a simply connec ed subdomain U⊂Gsuch ha
Gk⊂Uand (q)(z) (q+1)(z)= 0 o all z∈U. The exis ence o an app oxima ing
sequence {Pm}o polynomials on D(0, k +ε) in [6, Theo em 5] is he e gua an eed
by Runge’s heo em, which should be applied on V,Vbeing a simply connec ed
domain such ha Vis compac and Gk⊂V⊂V⊂U. We le he de ails o he
eade , which should ind no difficul y i he ollows s ep by s ep he p oo o he
ci ed e e ence. ////
ACKNOWLEDGEMENT
The au ho hanks he e e ees o some aluable commen s and sugges ions.
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