Sequences o di e en ial ope a o s: exponen ials,
hype cyclici y and equicon inui y
by
L. BERNAL–GONZ´
ALEZ and J.A. PRADO–TENDERO
Abs ac
In his pape , an eigen alue c i e ion o hype cyclici y due o he i s au ho is im-
p o ed. As a consequence, some new su icien condi ions o a sequence o in ini e o de
linea di e en ial ope a o s o be hype cyclic on he space o holomo phic unc ions on ce -
ain domains o
C
Na e shown. Mo eo e , se e al necessa y condi ions a e u nished. The
equicon inui y o a amily o ope a o s as be o e is also s udied, and i is e en cha ac e ized
i he domain is
C
N. The esul s ob ained ex end o imp o e ea lie wo k o se e al au ho s.
Key wo ds and ph ases: hype cyclic ope a o s and sequences, equicon inuous amily,
in ini e o de linea di e en ial ope a o , subexponen ial and exponen ial ype, eigen alue
c i e ion, o al subse , exponen ial unc ions, Runge domain, polydomain.
2000 Ma hema ics Subjec Classi ica ion: P ima y 47B38. Seconda y 30E10, 47A16,
47E05, 47F05.
This wo k has been suppo ed in pa by D.G.E.S. PB96–1348 and he Jun a de An-
daluc´ıa.
1 In oduc ion, no a ion and p elimina y esul s.
Th oughou his pape we deno e by
N
he se o posi i e in ege s, by
R
he eal
line, by
C
he ield o complex numbe s, and by
N
0 he se
N
0=
N
∪{0}. Le X, Y
be wo linea opological spaces, Ti:X→Y(i∈I:= an a bi a y index se ) a
amily o con inuous linea mappings, and x∈X. Then xis said o be hype cyclic
o uni e sal o (Ti) whene e i s o bi {Tix:i∈I}unde (Ti) is dense in Y.
The amily (Ti) is called hype cyclic whene e i has a hype cyclic ec o . No e
ha i (Ti) is hype cyclic hen i is no equicon inuous, bu he con e se is alse
in gene al. In he case I=
N
, i is clea ha , in o de ha a sequence (Tn) can
be hype cyclic, Ymus be sepa able. I T:X→Xis an ope a o (= con inuous
linea sel mapping) on X, hen a ec o x∈Xis said o be hype cyclic o Ti and
only i i is hype cyclic o he sequence (Tn) o i e a es o T, i.e., Tn=T◦T◦· · ·◦T
(n– old). The ope a o Tis hype cyclic when he e is a hype cyclic ec o o T.
The symbols HC(T) and HC((Ti)) will deno e, espec i ely, he se o hype cyclic
ec o s o an ope a o Tand o a amily Ti:X→Y(i∈I) o con inuous
1
linea mappings. In he las wo decades an ex ensi e li e a u e abou he opic o
hype cyclici y has been de eloped; a good su ey o he whole his o y is [G 1].
Le Gbe a nonemp y open subse o
C
N(N∈
N
). We say ha Gis a domain
when, in addi ion, i is connec ed. A domain G⊂
C
Nis said o be a Runge
domain (see [Ho ] o [K a]) i and only i each holomo phic unc ion on Gcan be
uni o mly app oxima ed by polynomials on compac subse s o G. No e ha , i
N= 1, hen Gis a Runge domain i and only i i is simply connec ed. By H(G)
we deno e, as usual, he F ´eche space o holomo phic unc ions on G, endowed
wi h he compac -open opology. Recall ha he amily {V(K, ε) : ε > 0, K is
a compac subse o G}is a neighbou hood basis o he o igin in H(G). He e
V(K, ε) := { ∈H(G) : || ||K< ε}. Fo A⊂
C
Nwe ha e deno ed ||g||A:=
sup{|g(z)|:z∈A}whene e gis a complex unc ion de ined on he se A.
G. Gode oy and J.H. Shapi o [GoS, Sec ion 5] p o ed in 1991 he ollowing
gene aliza ion o he classical app oxima ion heo ems by ansla es and de i a i es
o a single en i e unc ion due espec i ely o Bi kho [Bi ] and MacLane [Mac]:
I Tis an ope a o on he space H(
C
N) o en i e unc ions on
C
N ha commu es
wi h each o he ansla ion ope a o s τa(a∈
C
N) gi en by τa (z) = (z+a)
( ∈H(
C
N), z ∈
C
N), and is no a scala mul iple o he iden i y, hen HC(T) is
a dense Gδ-subse o H(
C
N); in addi ion, HC(T) con ains all nonze o unc ions o
a dense, T–in a ian , linea submani old o X:= H(
C
N). P. Bou don [Bou] and
D. He e o [He ] p o ed independen ly ha e e y hype cyclic ope a o Ton any
Banach space X(in ac , on any eal o complex locally con ex space X; see [Ans]
and [Bes]) has he same p ope y. The i s au ho o he p esen pape [Be4] has
ecen ly shown ha i Xand Ya e wo sepa able me izable linea opological
spaces and i Tn:X→Y(n∈
N
) is a sequence o con inuous linea mappings o
which he e is an inc easing sequence (nj) o posi i e in ege s wi h he p ope y
ha HC((Tmj)) is dense o e e y subsequence (mj) o (nj), hen HC((Tn)) ∪ {0}
con ains a dense linea submani old o X.
Gi en N∈
N
, deno e by Dj(1 ≤j≤N) complex pa ial di e 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(wi h D0
j=I= he iden i y ope a o o e e y j∈ {1, ..., N}), and |z|=
(|z1|2+· · · +|zN|2)1/2,zp=zp1
1· · · zpN
N,zw =z1w1+· · · +zNwNi z= (z1, ..., zN),
w= (w1, ..., wN). An en i e unc ion Φ(z) = P|p|≥0apzpis 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|
2
(z∈
C
N). Fo la e e e ences, we deno e by E he class o all en i e unc ions o
exponen ial ype. An en i e unc ion Φ is said o be 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 |Φ(z)| ≤ Aeε|z|
(z∈
C
N). E e y en i e unc ion o subexponen ial ype is ob iously in E. I is easy
o ealize (see, o ins ance, [Val], [Dic] o [Be3]) ha i G⊂
C
Nis a nonemp y
open subse and Φ is an en i e unc ion as abo e wi h subexponen ial ype, hen
he se ies Φ(D) = P|p|≥0apDpde ines an ope a o on H(G). I G=
C
N, he same
esul holds jus by assuming ha Φ is o exponen ial ype. So Φ(D) de ines, unde
he la e condi ions, an in ini e o de linea di e en ial ope a o wi h cons an
coe icien s. I is shown in [GoS] ha , gi en an ope a o Lon H(
C
N), hen L
commu es wi h e e y ansla ion ope a o τa(a∈
C
N) i and only i Lcommu es
wi h each Dk(1 ≤k≤N) i and only i L= Φ(D) o some en i e unc ion Φ in
E.
As a consequence o an eigen alue c i e ion o hype cyclici y [Be3, Theo em 7],
he i s au ho ob ained some ex ensions o Gode oy–Shapi o’s esul [Be3, Theo-
ems 8–9], his ime abou he hype cyclici y o a sequence o ope a o s (Φn(D)) de-
ined on he space o holomo phic unc ions on a Runge domain Go
C
N. Fu he -
mo e, condi ions abou he equicon inui y o a sequence (cnDn), whe e (cn)⊂
C
(no e ha his is he special case Φn(z) = cnzn), a e shown in [Be1] and [Be2] (see
also [Cal], when each cnis eplaced o a holomo phic uc ion cn(z)).
Ou aim in his pape is o p o ide wi h a mo e gene al eigen alue c i e ion and,
as a consequence, new su icien condi ions o he hype cyclici y o a sequence
o in ini e o de linea di e en ial ope a o s. In addi ion, necessa y condi ions
a e es ablished, and some special cases a e analyzed. Necessa y condi ions and
su icien condi ions o i s equicon inui y a e also u nished, and in pa icula we
cha ac e ize comple ely he equicon inui y in H(
C
N).
2 Eigen alues, exponen ials, hype cyclici y and equicon i-
nui y.
Likewise in [GoS, Sec ion 5] and [Be3, Theo ems 8–9], he key o he p oo o
hype cyclici y is o p o ide a good supply o eigen ec o s o he co esponding
ope a o s. Recall ha , in a linea opological space, a subse is said o be o al
whene e i s linea span is dense. I Tis an ope a o and eis an eigen ec o , hen
3
we deno e by λ(T, e) i s co esponding eigen alue. Nex , we s a e as a lemma he
ollowing a he gene al hype cyclici y c i e ion, which can be ound in [G 1].
Lemma 2.1 Assume ha Xis a Bai e opological ec o space, Yis a sepa able
me izable opological ec o space and Tn:X→Y(n∈
N
)a e con inuous linea
mappings. Suppose ha he e a e dense subse s X0o Xand Y0o Yand mappings
Sn:Y0→Xsuch ha
(a) o e e y x∈X0, he e exis s an inc easing sequence (nk) = {n1< n2< ...}
o posi i e in ege s wi h Tnkx→0 (k→ ∞),
(b) o e e y y∈Y0,(Sny)con e ges, and
(c) o e e y y∈Y0,Tn(Sny)→y(n→ ∞).
Then HC((Tn)) is esidual.
As no ed in [G 1, Rema k 2], i all he limi s in (b) a e ze o hen we may weaken
(a) o be o e e y x∈X0, he e exis s an inc easing sequence (nk) = {n1< n2<
...}o posi i e in ege s such ha (Tnkx)con e ges. Fu he mo e, he quan i ie
“∃(nk)” can be shi ed om (a) o (b) o (c).
Unde he same hypo hesis o Xand Y, i can be p o ed (see, o ins ance,
[Be2]) ha he ollowing condi ion is also su icien in o de ha HC((Tn)) be
esidual: he e exis dense subse s X0o Xand Y0o Ysa is ying ha o e e y
x∈X0and e e y y∈Y0 he e exis s an inc easing sequence (nk) o posi i e
in ege s and a sequence (xk)⊂Xsuch ha xk→0, Tnkx→0 and Tnkxnk→yas
k→ ∞.
By using he la e esul , he nex eigen alue c i e ion can be p o ed (see [Be3,
Theo em 7]): Le Xbe a sepa able F–space and (Tn) a sequence o ope a o s on
X. Assume ha he e a e wo o al subse s A,Bo Xsa is ying ha o e e y
pai o ini e subse s F1⊂ A and F2⊂ B he e is an inc easing sequence (nk) in
N
such ha e e y elemen in F1∪ F2is an eigen ec o o each Tnkin such a way
ha λ(Tnk, a)→0 (k→ ∞) o all a∈ F1and λ(Tnk, b)→ ∞ (k→ ∞) o all
b∈ F2. Then HC((Tn)) is esidual.
I we employ Lemma 2.1 (and he no e a e i ) ins ead o he jus men ioned
esul hen he ollowing eigen alue c i e ion can be ob ained. The p oo is le o
he in e es ed eade .
4
Theo em 2.2 Le Xbe a sepa able F–space and (Tn)be a sequence o ope a o s
on X. Assume ha he e a e wo o al subse s A,Bo Xsa is ying a leas one
o he ollowing condi ions:
(A) Fo e e y ini e subse F ⊂ A he e is an inc easing sequence (nk)in
N
such
ha e e y elemen in Fis an eigen ec o o each Tnkin such a way ha
λ(Tnk, a)→0 (k→ ∞) o all a∈ F. In addi ion, e e y elemen in Bis
an eigen ec o o each Tnin such a way ha o e e y b∈ B he sequence
(λ(Tn, b)) con e ges o a nonze o scala .
(B) Fo e e y ini e subse F ⊂ A he e is an inc easing sequence (nk)in
N
in such
a way ha o e e y a∈ F he sequence (λ(Tnk, a)) con e ges. In addi ion,
e e y elemen in Bis an eigen ec o o each Tnin such a way ha , o e e y
b∈ B,(λ(Tn, b)) → ∞ (n→ ∞).
(C) E e y elemen in Ais an eigen ec o o each Tnin such a way ha λ(Tn, a)→
0 (n→ ∞) o e e y a∈ A. In addi ion, o e e y ini e subse F ⊂ B
he e is an inc easing sequence (nk)in
N
such ha e e y elemen in Fis
an eigen ec o o each Tnkin such a way ha o e e y b∈ F he sequence
(λ(Tnk, b)) con e ges o a nonze o scala .
(D) E e y elemen in Ais an eigen ec o o each Tnin such a way ha o e e y
a∈ A he sequence (λ(Tn, a)) con e ges. In addi ion, o e e y ini e subse
F ⊂ B he e is an inc easing sequence (nk)in
N
such ha e e y elemen in F
is an eigen ec o o each Tnkin such a way ha (λ(Tnk, b)) → ∞ (k→ ∞)
o e e y b∈ F.
Then HC((Tn)) is esidual.
In o he o de o ideas, ecall ha Edeno es he class o en i e unc ions on
C
N
o exponen ial ype. We say ha a subse S⊂
C
Nis an E–unici y se whene e
he ollowing p ope y holds: i ∈ E and (z) = 0 o all z∈S hen ≡0.
No e ha , by he iden i y p inciple o holomo phic unc ions, i is an a bi a y
en i e unc ion anishing a Sand Sis a nonemp y open se (o e en jus a se
wi h a leas an accumula ion poin i N= 1) hen ≡0. This is no necessa y
o he class E; o ins ance, i N= 1 and χ:= lim sup →∞
log n( )
log >1, whe e
n( ) is he numbe o poin s o S∩ {|z| ≤ }, hen Sis an E–unici y se (e.g.,
S={n1/2:n∈
N
}, which gi es χ= 2). Indeed, i 6≡ 0, he la e condi ion
5

would imply ha he con e gence exponen o he sequence o ze os o is s ic ly
g ea e ha he g ow h o de o , which is clea ly impossible. The nex lemma will
be use ul la e . I s p oo is classical, bu we include i o he sake o comple eness.
I c∈
C
N hen we deno e ec(z) = exp(cz).
Lemma 2.3 I Sis an E–unici y se hen M(S) := {ec:c∈S}is o al in
H(
C
N).
P oo . Fix a unc ional L∈H(
C
N)∗(= he opological dual space o H(
C
N))
such ha L(ec) = 0 o all c∈S. Conside he Laplace ans o m ˜
Lo L(see
[Ho , p. 100]) gi en by ˜
L(z) = L(ez) (z∈
C
N). Then i is easy o show ha ˜
L
is an en i e unc ion on
C
No exponen ial ype which anishes a S. Since Sis
an E–unici y se , we ge ˜
L≡0. Then (Dp˜
L)(0) = 0 o all p∈
N
N
0. Bu i is
easy o show by induc ion ha (Dp˜
L)(0) = L(αp), whe e αp( ) = p( ∈
C
N). By
linea i y, L anishes a e e y polynomial, so L≡0 because he se o polynomials
is dense in H(
C
N). Summa izingly, i L( ) = 0 o all ∈M(S) hen L( ) = 0 o
all ∈H(
C
N). By he Hahn–Banach heo em, he linea span o M(S) is dense
in H(
C
N) o , equi alen ly, M(S) is o al.
Nex , we s a e he e eigh condi ions ha may o may no be sa is ied by a
sequence (Φn)⊂H(
C
N). Recall ha i Φ(z) = P|p|≥0apzp∈H(
C
N) and Φ is no
iden ically ze o, i s mul iplici y o he ze o a he o igin is m(Φ) = min{|p|:ap6=
0}. No e ha Φ(D)ec= Φ(c)ec o all c∈
C
N, so ecis an eigen ec o o Φ(D) wi h
eigen alue Φ(c).
(P) The e a e wo E–unici y se s A, B in
C
Nsuch ha o e e y pai o ini e
subse s F1⊂Aand F2⊂B he e exis s an inc easing sequence (nk)⊂
N
wi h Φnk(a)→0 (k→ ∞) o all a∈F1and Φnk(b)→ ∞ (k→ ∞) o all
b∈F2.
(Q) The e is an E–unici y se Bin
C
Nsuch ha o e e y ini e subse F⊂B
he e exis s an inc easing sequence (nk)⊂
N
wi h m(Φnk)→ ∞ (k→ ∞)
and Φnk(b)→ ∞ (k→ ∞) o all b∈F.
(R) The e a e wo E–unici y se s A, B in
C
Nsuch ha o e e y ini e subse
F⊂A he e exis s an inc easing sequence (nk)⊂
N
wi h Φnk(a)→0 (k→ ∞)
o all a∈F, and o each b∈B he sequence (Φn(b)) con e ges o a nonze o
complex numbe .
6
(S) The e is an E–unici y se Bin
C
Nsuch ha o each b∈B he sequence
(Φn(b)) con e ges o a nonze o complex numbe , and he e exis s an inc easing
sequence (nk)⊂
N
wi h m(Φnk)→ ∞ (k→ ∞).
(T) The e a e wo E–unici y se s A, B in
C
Nsuch ha o e e y ini e subse
F⊂A he e exis s an inc easing sequence (nk)⊂
N
sa is ying ha o e e y
a∈F he sequence (Φnk(a)) con e ges. In addi ion, Φn(b)→ ∞ (n→ ∞)
o e e y b∈B.
(U) The e a e wo E–unici y se s A, B in
C
Nsuch ha Φn(a)→0 (n→ ∞) o
all a∈A, and o each ini e subse F⊂B he e exis s an inc easing sequence
(nk)⊂
N
sa is ying ha o e e y b∈F he sequence (Φnk(b)) con e ges o a
nonze o complex numbe .
(V) The e is an E–unici y se Bin
C
Nsuch ha o each ini e subse F⊂B
he e exis s an inc easing sequence (nk)⊂
N
sa is ying ha o e e y b∈F
he sequence (Φnk(b)) con e ges o a nonze o complex numbe . In addi ion,
m(Φn)→ ∞ (n→ ∞).
(W) The e a e wo E–unici y se s A, B in
C
Nsuch ha o e e y a∈A he
sequence (Φn(a)) con e ges, and o e e y ini e subse F⊂B he e is an
inc easing sequence (nk)⊂
N
wi h Φnk(b)→ ∞ (k→ ∞) o all b∈F.
We a e now eady o s a e ou nex esul . In he emaining o his pape , Φ and
Φi(i∈I:= an a bi a y index se ) will deno e en i e unc ions o subexponen ial
ype i G6=CNand o exponen ial ype i G=
C
N,Gbeing a gi en domain in
C
N. Thus, he ope a o s Φ(D), Φi(D) (i∈I) a e well de ined on H(G).
Theo em 2.4 Suppose ha Gis a Runge domain o
C
Nand ha (Φn)sa is ies
a leas one o he condi ions (P)–(W). Then HC((Φn(D))) is esidual in H(G).
P oo . Recall ha , by Lemma 2.3, he se M(S) is o al in H(
C
N) (hence
in H(G), because Gis Runge) whene e Sis an E–unici y se . Recall also ha
he se {zp:p∈
N
N
o}is o al in H(G), because ha se spans {polynomials}.
Take X=H(G) and Tn= Φn(D) (n∈
N
). Then: Apply he esul men ioned
jus be o e Theo em 2.2 on A=M(A), B=M(B) i (Φn) sa is ies (P), and on
A={zp:p∈
N
N
0},B=M(B) i (Φn) sa is ies (Q). Apply condi ion (A) o
Theo em 2.2 on A=M(A), B=M(B) i (Φn) sa is ies (R), and on A={zp:
p∈
N
N
0},B=M(B) i (Φn) sa is ies (S). Apply condi ion (B) o Theo em 2.2
7
on A=M(A), B=M(B) i (Φn) sa is ies (T). Apply condi ion (C) o Theo em
2.2 on A=M(A), B=M(B) i (Φn) sa is ies (U), and on A={zp:p∈
N
N
0},
B=M(B) i (Φn) sa is ies (V). Finally, apply condi ion (D) o Theo em 2.2 on
A=M(A), B=M(B) i (Φn) sa is ies (W).
Le us u nish se e al examples ha illus a e Theo em 2.4. The eade will
ealize ha none o he examples below can be de i ed om Theo ems 8, 9 o
[Be3]. Bu be o e his we should ix some subse s. Conside S={n1/2:n∈
N
}
and le ( j) be any sequence o posi i e eal numbe s such ha he plane disks
{|z−j1/2|< j}(j∈
N
) be pai wise disjoin , o ins ance, j= 1/6j. De ine he
compac s se s Kn:= (Ln∪S)∩In(n∈
N
), whe e
In:= [−n, n]×[−n, n]
and
Ln:=
C
[((0,+∞)×(−1/n, 0)) ∪
∞
[
j=1
{|z−j1/2|< j/n}].
I is easy o see ha each Knhas connec ed complemen . De ine he unc ions
n, gn:Kn→
C
(n∈
N
) as
n(z) = 


1 (z∈Ln∩In)
n(z∈S∩In)
and
gn(z) = 


1 (z∈Ln∩In)
0 (z∈S∩In).
I is clea ha e e y nand e e y gnis holomo phic on some open subse con ain-
ing Knand depending on n. Then Runge’s heo em gua an ees he exis ence o
polynomials Pn, Qnsa is ying
||Pn− n||Kn<1/n and ||Qn−gn||Kn<1/n (n∈
N
).
Since Ln∩In(S∩In) g ows up o
C
S(up o S, espec i ely) as n ends o in ini y,
he la e wo inequali ies lead us o he ollowing ac s o poin con e gence:
Pn→1 on
C
S,Pn→ ∞ on S,Qn→1 on
C
Sand Qn→0 on Sas n→ ∞.
EXAMPLE 1. The e is a esidual se o en i e unc ions on
C
such ha each
en i e unc ion can be locally uni o mly app oxima ed by en i e unc ions o he
o m n
X
j=0
Ajn (j)(n∈
N
),
8
whe e Ann = 1 and
Ajn = (−1)n−jX
1≤i1<i2<···<in−j≤n
(i1· · · in−j)1/2(0 ≤j≤n−1).
Indeed, i su ices o apply he la e heo em wi h condi ion (P) o (T) on A=S,
B=
C
S, Φn(z) =
n
Y
j=1
(z−j1/2) (n∈
N
) (use Ca dano–Vie a’s ela ions).
EXAMPLE 2. The se HC((Pn(D))) is esidual in H(
C
) because Theo em 2.4 can
be applied wi h condi ion (T) o (W) on A=
C
S,B=S.
EXAMPLE 3. The se HC((Qn(D))) is esidual in H(
C
) because Theo em 2.4
can be applied wi h condi ion (R) o (U) on A=S,B=
C
S.
Analogous p ope ies o (P)–(W) ega ding he densely he edi a y hype cyclic-
i y o (Φn(D)) can be o mula ed as in [Be4, Sec ion 3]. This would yield su icien
condi ions o he exis ence o dense (Φn(D))–hype cyclic linea submani olds in
H(G).
In his pape , Bi kho [Bi ] essen ially p o ed ha gi en an unbounded sequence
(an)⊂
C
he e exis s an en i e unc ion in
C
such ha he se o ansla es
{ (z+an) : n∈
N
}is dense in H(
C
), i.e., he sequence (τan) is hype cyclic (as a
ma e o ac , he sequence (an) depended on he pa icula en i e unc ion o be
app oxima ed; in [Luh] his dependence is d opped). His cons uc i e p oo can be
adap ed o
C
N: see, o ins ance, [Abe] and [AbZ]; see also [A G] o co esponding
esul s o ha monic unc ions on
R
N. As a quick applica ion o he la e heo em,
we will ob ain his Bi kho heo em in se e al a iables.
Theo em 2.5 Assume ha S⊂
C
N. Then he ollowing condi ions a e equi alen :
(a) Sis unbounded.
(b) The amily o ope a o s (τa)a∈Sis hype cyclic on H(
C
N).
(c) HC((τa)a∈S)is esidual in H(
C
N).
(d) (τa)a∈Sis no equicon inuous on H(
C
N).
P oo . The implica ions (c) ⇒(b) ⇒(d) a e i ial. I Sis bounded, ake
M∈(0,+∞) wi h |a| ≤ M o all a∈S. Gi en a basic neighbou hood V(K, ε)
o he o igin in H(
C
N), i is clea ha
[
a∈S
τa(V(L, δ)) ⊂V(K, ε),
9
en i e unc ion wi h subexponen ial ype, so pa (a) o Theo em 2.9 yields he
desi ed esul .
Fo G=
C
Nwe a e able o cha ac e ize he equicon inuous amilies o di e en-
ial ope a o s.
Theo em 2.11 The amily o ope a o s {Φi(D) : i∈I}is equicon inuous on
H(
C
N)i and only i (Φi)admi s a majo an en i e unc ion o exponen ial ype.
P oo . The pa “only i ” is due o Theo em 2.9(b). As o he con e se, we can
ollow s ep by s ep he p oo o pa (a) o Theo em 2.9 wi h he sole excep ion
ha we may choose he polycycle γ a enough om he compac se K(so µcan
be choosen as la ge as desi ed) in such a way ha
sup
|p|>0, i∈I
(p!|cpi|)1/|p|≤µ/2.
The cons an Mmay be choosen as M= max {1,supi∈I|c0i|}. The p oo is in-
ished.
In [Be2, Theo em 1] i has been es ablished ha i G⊂
C
is a simply con-
nec ed domain and (cn) is a complex sequence wi h R(G)≤lim sup
n→∞ (n!|cn|)1/n
hen HC((cnDn)) is esidual in H(G). A sligh gene aliza ion can be ob ained in
he N–dimensional case. The p oo is e y simila o he 1–dimensional one, so we
omi i .
Theo em 2.12 Assume ha G⊂
C
Nis a Runge domain and ha (p(n)) is a
sequence o mul i–indexes wi h |p(n)|→∞(n→ ∞). I (cn)is a complex sequence
wi h
R(G)≤lim sup
n→∞ (p(n)! |cn|)1/|p(n)|
hen he se HC((cnDp(n))) is esidual in H(G).
As a consequence o Theo ems 2.11, 2.12 we can ge a cha ac e iza ion o
equicon inui y and hype cyclici y o he same sequence in H(
C
N). This is achie ed
in he nex esul , which in u n is an N–dimensional ex ension o [Be2, Theo em
4] (see also [Be1]).
Theo em 2.13 Assume ha (cn)is a complex sequence and ha (p(n)) is a
sequence o nonze o mul i–indexes such ha |p(n)|→∞(n→ ∞). Then he
ollowing p ope ies a e equi alen :
16

(a) The sequence ((p(n)! |cn|)1/|p(n)|)is bounded.
(b) The e is no hype cyclic en i e unc ion o (cnDp(n)).
(c) The se HC((cnDp(n))) is no esidual in H(
C
N).
(d) The sequence (cnDp(n))is equicon inuous on H(
C
N).
P oo . I is e iden ha (b) implies (c) and ha (d) implies (b). Since R(
C
N) =
+∞, we ob ain om Theo em 2.12 ha (c) implies (a). Assume ha (a) holds.
Then we can apply Theo em 2.11 wi h I=
N
and Φn(z) = cnzp(n). Indeed, he e
is a cons an Mwi h p(n)!|cn| ≤ M|p(n)| o all n∈
N
, hence he unc ion
Φ(z) =
∞
X
n=1
M|p(n)|
p(n)! zp(n)(z∈
C
N)
is a majo an en i e unc ion o (Φn) wi h exponen ial ype. Then (d) is ue and
he p oo is inished.
We poin ou ha in [G 2, Co olla y o Theo em 4] he pa abou hype cyclici y
o [Be2, Theo em 4] is ex ended o he case N= 1 o sequences o weigh ed
pseudo-shi s in he space H(
C
).
The pa “only i ” o [Be2, Theo em 3] is able o be ex ended in he same way
o he N–dimensional case, as he ollowing heo em shows.
Theo em 2.14 Le G=G1× · · · × GN⊂
C
Nbe a polydomain wi h Gj6=
C
(j= 1, ..., N). Assume ha (cn)is a complex sequence and ha (p(n)) is a
sequence o nonze o mul i–indexes such ha he sequence o ope a o s (cnDp(n))is
equicon inuous on H(G). Then
lim
n→∞ (p(n)! |cn|)1/|p(n)|= 0.
P oo . Conside he numbe α:= lim supn→∞(p(n)!|cn|)1/|p(n)|. By he way o
con adic ion, assume ha α > 0. Fix a poin a= (a1, ..., aN)∈G. Then aj∈Gj
and he e exis poin s bj∈
C
Gj(j= 1, ..., N) such ha |aj−bj|= in {|aj− |:
∈
C
Gj}. Deno e R= min {|aj−bj|:j= 1, ..., n}>0. Fix ∈(0, R) wi h
R− < α. Pu K=D(a, ). Then Kis a compac subse o G. Le Lbe any
compac subse o Gand δa posi i e numbe . Le m > 0 be so small ha
m
(in {|zj−bj|:j∈ {1, ..., N}, z = (z1, ..., zn)∈L∪K})N< δ.
17
Conside he unc ion
(z) = m
QN
j=1(zj−bj).
Then ∈H(G) and, in addi ion, belongs o V(δ, L). Fu he mo e, o z=
(z1, ..., zN)∈G,
|(Tn )(z)|=p(n)! m|cn|
QN
j=1 |zj−bj|1+pj(n),
whe e p(n)=(p1(n), ..., pN(n)) and Tn=cnDp(n). Since in {| −bj|:| −aj|<
} ≥ R− o e e y j∈ {1, ..., N}, we ge
sup {|(Tn )(z)|:z∈K} ≤ p(n)!|cn|m
(R− )N+|p(n)|=m
(R− )N·p(n)!|cn|
(R− )|p(n)|.
Bu p(nk)!|cnk|
(R− )|p(nk)|→ ∞ (k→ ∞) o some inc easing sequence (nk)⊂
N
, because
α > R − . Hence sup{|Tn (z)|:z∈K}=∞. The e o e
[
n∈
N
Tn(V(δ, L)) 6⊂ V(1, K),
which implies ha (Tn) is no equicon inuous. The p oo is inished.
Ou inal esul comes back o hype cyclici y and looks sligh ly di e en om
he o he s. I pu s he emphasis on he i s nonze o Taylo coe icien o each Φn.
This ime he se ing is he complex plane
C
. Obse e ha MacLane’s heo em is
again eco e ed i we choose Φn(z) = zn o each n∈
N
.
Theo em 2.15 Assume ha (Φn(z) =
∞
X
j=0
cjnzj)is a sequence o nonze o en i e
unc ions and deno e p(n) := m(Φn) (n∈
N
). Assume ha he ollowing h ee
condi ions a e ul illed:
(a) p(n)→ ∞ as n→ ∞.
(b) p(n)|cp(n),n|k/p(n)→ ∞ as n→ ∞ o e e y k∈
N
.
(c) Each sequence {cj+p(n),n :n∈
N
}(j∈
N
)is bounded.
Then he se HC((Φn(D))) is esidual in H(G) o any simply connec ed domain
G⊂
C
.
P oo . We a e ying o apply Lemma 2.1 wi h X=H(G) = Y,X0=
{polynomials}=Y0and Tn= Φn(D) (n∈
N
). I Pis a polynomial hen by
18
(a) he e exis s n0∈
N
wi h p(n)>deg ee (P) o all n≥n0, hence DjP= 0 o
all j≥p(n) (n≥n0). The e o e TnP= 0 e en ually and condi ion (a) o Lemma
2.1 is sa is ied. Now ix mand nin
N
and y o sol e he equa ion Tn =zm.
Obse e ha Tn= Ψn(D)◦Dp(n), whe e Ψn(z) = P∞
j=0 ajnzjand ajn =cj+p(n),n,
so a0n6= 0 o all n∈
N
. Conside he equa ion
Ψn(D)g=zm,(1)
whe e gis a polynomial o deg ee no g ea e han m, say, g(z) = Pm
k=0 bknzk. I
is easy o see ha such a polynomial solu ion exis s. Indeed, (1) is equi alen o
m
X
j=0
ajn(
m
X
k=0
bknzk)(j)=zm,
which in u n is he same as he sys em



Pm
j=kaj−k,nbjn ·j!
k!= 0 (k= 0,1, ..., m −1)
a0nbmn = 1.
This is a ecu en squa e sys em wi h de e minan am+1
0n6= 0, so i has a unique
solu ion (b0n, ..., bmn) and C ame ’s ule yields
bkn =1
am+1
0n
·
m
X
j=1
Pjkm(a1n, ..., amn)aj
0n(2)
o k∈ {0,1, ..., m}, whe e Pjkm (j= 1, ..., m) a e polynomials o mcomplex
a iables no depending on n. F om (c), he e is a ini e posi i e cons an M,
which does no depend on n, such ha
|Pjkm(a1n, ..., amn)| ≤ M(3)
o all k∈ {0,1, ..., m}and all j∈ {1, ..., m}. Hence a solu ion o Tn =zmis
(z) = n(z) =
m
X
k=0
bkn
zk+p(n)
(k+p(n))! (n∈
N
),
whe e bkn is gi en by (2). Le us ix R > 1. Then om (3) we ob ain o |z| ≤ R
ha
| n(z)| ≤ (m+ 1)
m
X
j=1
MRm
|a0n|m+1−j·Rp(n)
p(n)! →0 (n→ ∞)
since (b) and S i ling’s o mula leads us o
(p(n)! |a0n|m+1−j)1/p(n)→ ∞ (n→ ∞),
19
so he e ms o he la e sequence a e e en ually g ea e han, o ins ance, 1/2R.
The e o e ( n) ends o ze o in H(G). The p oo o he case m= 0 is easie and
le o he eade . De ine
Sn(zm) := n(z) (m∈
N
0;n∈
N
)
and ex end Sn o Y0by linea i y. Then i is clea ha SnP→0 (n→ ∞) and
Tn(SnP) = P→Pas n→ ∞. Consequen ly, condi ions (b) and (c) in Lemma
2.1 a e also ul illed, as equi ed.
Fo ins ance, he e is an en i e unc ion in
C
wi h he p ope y ha any en i e
unc ion can be locally uni o mly app oxima ed by unc ions o he o m
cn( (n)+ (n+1)) (n∈
N
),
whe e cn=n−n/(log n)1/2. Indeed, he sequence {Φn(z) = cnzn(1 + z)}sa is ies
all hypo heses o he la e heo em, because (cn) is bounded, p(n)→ ∞ and
p(n)·n−kn/(p(n) (log n)1/2)→ ∞ (n→ ∞) o all k∈
N
, whe e p(n)≡nhe e. No e
ha his example shows ha Theo em 2.15 is no included in Theo em 2.4: in
ac , Φn(z)→0 as n→ ∞ o all z∈
C
, hence he E–unici y se Bis no a ailable
in o de o apply he men ioned heo em.
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20
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21

[Luh] W. Luh, On uni e sal unc ions, Colloq. Ma h. Soc. J´anos Bolyai 19 (1976),
503–511.
[Mac] G. R. MacLane, Sequences o de i a i es and no mal amilies, J. Analyse
Ma h. 2(1952), 72–87.
[Rud] W. Rudin, Real and Complex Analysis, 2nd. ed., Ta a McG aw–Hill, Fa id-
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[Val] G. Vali on, Su les solu ions des ´equa ions di e en ielles line´ai es d’o d e
in inie e `a coe icien s cons an s, Ann. ´
Ecole No m. (3) 46 (1929), 25–53.
LUIS BERNAL–GONZ ´
ALEZ JOS´
E ANTONIO PRADO–TENDERO
DEPARTAMENTO DE AN ´
ALISIS MATEM ´
ATICO DEPARTAMENTO DE AN ´
ALISIS MATEM ´
ATICO
FACULTAD DE MATEM ´
ATICAS, APDO. 1160 FACULTAD DE MATEM´
ATICAS, APDO. 1160
AVENIDA REINA MERCEDES AVENIDA REINA MERCEDES
41080–SEVILLA, SPAIN 41080–SEVILLA, SPAIN
E–mail: lb[email p o ec ed] E–mail: [email p o ec ed]
22