Ope a o s wi h dense images e e ywhe e*
by
L. BERNAL-GONZ´
ALEZ** and M.C. CALDER´
ON-MORENO**
Abs ac
In his pape , he au ho s in oduce he dense-image ope a o s Tas hose wi h a
wild beha iou nea o he bounda y o a domain G, ia ce ain subse s. The ela ion-
ship wi h o he kinds o ope a o s wi h wild beha iou is s udied, p o ing ha he new
concep gene alizes he ea lie o omnip esen , bu he e is no good ela ionship wi h
he s ongly omnip esen ope a o s. We ob ain, among o he esul s, ha he ollowing
kinds o ope a o s a e dense-image: on o linea ope a o s; ope a o s wi h local dense
ange sa is ying so condi ions; Vol e a complex in eg al ope a o s plus in ini e o de
di e en ial ope a o s, mul iplica ion ope a o s. In addi ion, holomo phic sel mappings
and en i e unc ions gene a ing dense-image igh o le composi ion ope a o s a e
comple ely cha ac e ized.
Key wo ds and ph ases: dense-image ope a o , omnip esen ope a o , s ongly om-
nip esen ope a o , di e en ial ope a o , an idi e en ial ope a o , in eg al ope a o ,
holomo phic unc ion, composi ion ope a o , le -composi ion ope a o , mul iplica ion
ope a o , local dense ange, esidual se , non- ela i ely compac se .
2000 Ma hema ics Subjec Classi ica ion: P ima y 30E10. Seconda y 30H05,
46E10, 47B38, 47E05, 47G10.
*This wo k has been pa ially suppo ed by DGES g an BFM2001–2717 and he
Jun a de Andaluc´ıa.
**Depa amen o de An´alisis Ma em´a ico, Facul ad de Ma em´a icas, A da. Reina
Me cedes, Apdo. 1160, 41080 Se illa, Spain. E–mails: lb[email p o ec ed], [email p o ec ed]
1 In oduc ion and no a ion
In wha ollows Nwill be he se o posi i e in ege s, N0=N∪ {0},C
is he complex plane, C∞is he ex ended complex plane C∪ {∞},B(a, )
is he euclidean open ball wi h cen e aand adius (a∈C, > 0). The
open uni ball is D=B(0,1). Gwill s and o a nonemp y open subse o
C(o a complex domain, ha is, a nonemp y connec ed open subse o C),
∂G is he bounda y o Gin C∞. We deno e O(∂G) = {V⊂C∞:Vis
open and V∩∂G 6=∅}. I A⊂C hen A ep esen s he closu e o Ain C∞,
and k kA:= supz∈A| (z)|, whe e is a complex unc ion de ined on A, and
LT(A) := {a ine linea ans o ma ions τ(z) = az +bsuch ha τ(D)⊂A}.
H(G) deno es, as usual, he F ´eche space o holomo phic unc ions on G,
endowed wi h he opology τo local uni o m con e gence in G. In pa icula ,
H(G) is a Bai e space. Le K(G) be he amily o compac subse s o G
1
and K1(G) he amily o compac subse s Ko Gsuch ha each connec ed
componen o C∞ Kcon ains a leas one connec ed componen o C∞ G.
I is known ha he amily
{D( , K, ε) : ∈H(G), K ∈ K1(G), ε > 0},
whe e D( , K, ε) = {g∈H(G) : |g(z)− (z)|< ε o all z∈K}, is a basis
o τ. An ope a o on H(G) is a con inuous sel -mapping T:H(G)→H(G),
no necessa ily linea .
As a consequence o a well-known esul abou in e pola ion [15, Theo em
15.13], a unc ion ∈H(G) can be ound in such a way ha (A) is dense in
C, whe e Ais a p e ixed non- ela i ely compac subse o G(see also [7], [8,
Chap e IV], [11]). The i s au ho [2] has p o ed he ac ha , gi en A⊂G,
he se o unc ions ∈H(G) such ha (j)(A) is dense in C o all ∈N0
is esidual in H(G) i and only i Ais no ela i ely compac in G.
In 1998, he second au ho [6] ex ended hese ac s by conside ing ope a o s
each o hem being he sum o an in ini e o de di e en ial ope a o and an
in eg al ope a o . Gi en an ope a o Ton H(G) and a subse A⊂G, we
deno e M(T, A) = {g∈H(G) : (Tg)(A) is dense in C}. Then he main esul
o [6] eads as ollows.
Theo em 1.1 Le G⊂Cbe a simply connec ed domain, A⊂G, b ∈G,
Φ(z) = P∞
j=0 ajzjan en i e unc ion o subexponen ial ype i G6=Cand o
exponen ial ype i G=C, and le ϕ:G×G→Cbe an analy ic unc ion wi h
espec o bo h a iables. Conside he linea ope a o Ton H(G)de ined by
T (z) = Zz
bϕ(z, ) ( )d (z∈G)
whe e he in eg al is aken along any ec i iable cu e in Gjoining b o z. Then
he ollowing p ope ies a e equi alen :
1. Ei he
(a) Φ(z)≡0on Gand o e e y compac subse Lo G he e exis
a∈A Land z∈Gsuch ha ϕ(a, z)6= 0, o
(b) Φ(z)6≡ 0on Gand Ais no ela i ely compac in G.
2. M(T+ Φ(D), A)is esidual in G.
3. M(T+ Φ(D), A)6=∅.
We ha e deno ed Φ(D) = P∞
j=0 ajDj,Dis he di e en ia ion ope a o
D = 0,D0=I= he iden i y ope a o and Dj+1 =D◦Dj. I is easy o see
ha Φ(D) is a well-de ined ope a o unde he condi ions o Theo em 1.1.
2
By ollowing ano he poin o iew, h ough “modi ied” clus e se s, ope-
a o s on H(G) wi h a wild beha iou nea he bounda y can be s udied.
Inspi ed by he no ion o “mons e ” in oduced and de eloped by W. Luh [12]
and K.G. G osse-E dmann [10] (see also [13], [14] and [16]), he au ho s ha e
ecen ly p esen ed he concep o “T-mons e ” and i s associa ed no ion o
“s ongly omnip esen ope a o ” in he nex way.
Gi en an ope a o Ton H(G), we say ha a unc ion ∈H(G) is a T-
mons e i and only i o each g∈H(D) and each ∈∂G he e exis wo
sequences {an}nand {bn}nin Csuch ha anz+bn→ as n→ ∞ uni o mly
on D,anz+bn∈G( o all n∈Nand all z∈D) and (T )(anz+bn)→g(z)
as n→ ∞ locally uni o mly in D.
We say ha Tis s ongly omnip esen whene e each subse U(T, g, ε, , V ) :=
{ ∈H(G) : he e exis s τ∈LT(V∩G) such ha k(T )◦τ−gk D< ε}
(g∈H(D), ε > 0, ∈(0,1), and V∈O(∂G)) is dense in H(G).
In [3] he au ho s p o ed ha :
(1) E e y s ongly omnip esen ope a o Tis omnip esen [1], i.e., he se
R(T, V, W) := { ∈H(G) : exis s z∈V∩Gsuch ha (T )(z)∈W}is
dense in H(G), o e e y V∈O(∂G) and e e y non-emp y open subse
W⊂C.
(2) Tis s ongly omnip esen i and only i he se M(T) o T-mons e s is
esidual.
See [3–5] o examples o s ongly omnip esen ope a o s.
Ou aim in his pape is:
(i) To in oduce a new ype o mapping, namely, he “dense-image ope a-
o s”, wi h wild beha io nea he bounda y o a domain ia he image
o plane se s, and o s udy i s possible ela ionship wi h omnip esen and
s ongly omnip esen ope a o s.
(ii) To cons uc new dense-image ope a o s om o he ones and o gi e
su icien condi ions o an ope a o o be dense-image.
(iii) To gi e se e al conc e e examples o dense-image ope a o s.
We a e going o de elop each o hese poin s in each o he ollowing h ee
sec ions.
2 Wha is a dense-image ope a o ?
Assume ha Tis an ope a o on H(G). We say ha Tis a dense-image
ope a o (DI-ope a o ) i and only i M(T, A) is esidual in H(G), o e e y
non- ela i ely compac subse A⊂G.
3
Rema k 2.1 In [6] i is p o ed ha he se M(T, A) is always a Gδ-subse , in
ac , i can be w i en as a coun able in e sec ion o se s HA(T, δ, w) := {g∈
H(G) : he e exis s a∈Asuch ha |(Tg)(a)−w|< δ}, whe e δ > 0 and
w∈C. So M(T, A) is esidual i and only i i is dense in H(G).
Rema k 2.2 We di ec ly demand A o be non- ela i ely compac because
o he wise M(T, A) is emp y due o he con inui y o T. As we can see in
Theo em 1.1 o Example 2.5, he con e se is no ue.
Rema k 2.3 I is ob ious ha o each subse Ao Gwhich is no ela i ely
compac he e exis a poin ∈A∩∂G and a sequence {an}n⊂Awi h an→
as n→ ∞. Fu he mo e, M(T, A)⊂M(T, B) whene e A⊂B⊂G.
F om he las ema k, we can easily ob ain an al e na i e de ini ion o
DI-ope a o s.
Theo em 2.4 Le Tbe an ope a o on H(G). Then he ollowing wo condi-
ions a e equi alen :
(1) Tis a DI-ope a o .
(2) Fo each poin ∈∂G and each sequence {an}n⊂Gwi h an→ as
n→ ∞ he se M(T, {an}n)is dense in H(G).
A i s example o DI-ope a o is, by Theo em 1.1, any nonze o sum o di e -
en ial and an idi e en ial in ini e o de ope a o s and in pa icula all nonze o
ope a o s Φ(D) and Ψ(D−1
b) (hence he iden i y ope a o I) a e dense-image
and s ongly omnip esen (see [3]). He e Ψ is any non-ze o unc ion ha is
holomo phic in a neighbou hood o he o igin, b∈G, Ψ(D−1
b) = P∞
j=0 bjD−j
b,
D0
b=Iand, o each j∈N,D−j
b ( ∈H(G)) deno es he unique j-
an ide i a i e Fo such ha F(k)(b) = 0 (k∈ {0,1, . . . , j −1}). No e
ha Ψ(D−1
b) can be easily exp essed as an in eg al ope a o Tas in Theo-
em 1.1, o ce ain ϕ(z, ) depending only on he di e ence z− . The ways
hese wo p ope ies (“dense-image” and “s ongly omnip esen ”) ha e been
ob ained a e e y simila and one could belie e ha bo h p ope ies a e he
same. This is no ue, howe e , as we a e going o see in he nex examples.
Example 2.5 I {zn}nis a sequence in D {0}such ha P∞
n=1(1−|zn|)<+∞
hen i s associa ed Blaschke p oduc is
B(z) =
∞
Y
1
|zn|
zn
zn−z
1−znz.
Then B(z) de ines a unc ion which is holomo phic in D. In addi ion, i
E⊂∂Dis he se o accumula ion poin s o {zn}n, hen B(z) ex ends o
4
be holomo phic on C (E∪ {zn−1:n∈N}) (see [9, Theo em 6.1]). I we
choose zn= 1 −1
n2(n≥2) hen E={1}and B∈H(C [{1}∪{1 + 1
n2−1:
n≥2}]). Since each ac o has modulus 1 on ∂Dwe ha e ha B(z)6= 0 o
all z∈Γ := (∂D) {1}(in ac , |B|= 1 on Γ).
Summa izing, we ha e ob ained a unc ion B(z) sa is ying:
(1) B∈H(D),
(2) B(z) = 0 i and only i z∈ {1−1
n2:n≥2},
(3) he e exis s a dense subse Γ ⊂∂Dsuch ha o e e y ∈Γ he e exis s
limz→ B(z)∈C {0}.
Conside he mapping
T:H(D)→H(D)
7→ T (z) = B(z) (z).
Then Tis a s ongly omnip esen ope a o (see [4, Theo em 3.7]) bu no a
DI-ope a o . Indeed, i we ake A={1−1
n2:n≥2} hen Ais no ela i ely
compac in Dand M(T, A) = ∅because T (A) = {0} o all ∈H(D).
The la e is a linea example. Le us gi e an addi ional one. I we compa e
Co olla y 4.2 o his pape wi h Co olla y 3.2 o [4], we ob ain ha each compo-
si ion ope a o Cϕon H(C) wi h ϕ anscenden is also s ongly omnip esen
bu no dense-image.
Example 2.6 In [4], i is shown ha o each en i e unc ion ϕ he le -
composi ion ope a o Lϕ:H(G)→H(G) gi en by Lϕ =ϕ◦ is no
s ongly omnip esen i ϕis no on o, so by Theo em 4.3 (see Sec ion 4) o
each en i e unc ion ϕwhich is non-cons an and non-on o he mapping Lϕis
a DI-ope a o which is no s ongly omnip esen . In ac , i ϕis no on o hen
M(Lϕ) is emp y, so he e exis DI-ope a o s wi hou any mons e .
The abo e example is always non-linea , so he ques ion may be asked,
is e e y linea DI-ope a o s ongly omnip esen ? We poin ou he e ha
he e a e linea omnip esen ope a o s which a e no s ongly omnip esen [4,
Example 3.6].
Le us see now he ela ionship be ween DI-ope a o s and omnip esen op-
e a o s.
P oposi ion 2.7 Le ∈∂G and le {an}nbe a sequence in Gwi h an→
as n→ ∞. Then M(T, {an}n)⊂R(T, V, W) o e e y non-emp y open subse
Wo Cand e e y open se Vwi h ∈V.
5

P oo . Fix a non-emp y open subse W⊂Cand an open se Vwi h ∈V.
Since an→ (n→ ∞), he e exis s n0∈Nsuch ha an∈V∩G o all
n≥n0. Le be an elemen o M(T, {an}n). Then
W∩(T ){an:n≥n0} 6=∅
because (T ){an:n≥n0}is dense in Cand Wis open. So he e exis s
n1∈Nsuch ha an1∈G∩Vand T (an1)∈W, whence
∈R(T, V, W).
The p oo is inished. ♦
In pa icula , i o each poin o a dense subse Γ o ∂G he e exis s a
sequence {an} ⊂ Gwi h an→ as n→ ∞ and M(T, {an}n) dense in H(G)
we ha e ha Tis omnip esen . F om he e we ge he nex co olla y:
Co olla y 2.8 E e y DI-ope a o is omnip esen .
Obse e ha , by Example 2.5, he con e se is no ue. In ac , i is easy
o check ha he example p o ided in [4, Example 3.6] is no dense-image
ei he . Hence, by asse ion (1) in he In oduc ion, we ha e ha linea om-
nip esen ope a o s s ic ly include bo h dense-image ope a o s and s ongly
omnip esen ope a o s. Howe e , as in Theo em 2.4, we can cha ac e ize he
omnip esen ope a o s in e ms o sequences nea he bounda y.
Theo em 2.9 The ollowing wo condi ions a e equi alen :
(1) Tis an omnip esen ope a o .
(2) The e is a dense subse o Γ⊂∂G such ha o e e y ∈Γ he e exis s
a sequence {an}n⊂Gwi h an→ as n→ ∞ and M(T, {an}n)dense in
H(G).
P oo . (2) ⇒(1) is done. We need o p o e (1) ⇒(2).
Fix ∈Γ, w∈Cand δ > 0, and o each n∈Nconside he open ball
Vn:= B( , 1
n). Then R(T, Vn, B(w, δ)) ∩D(h, K, ε)6=∅( o all h∈H(G), all
K∈ K(G) and all ε > 0), by (1). So, we can ind a unc ion ∈D(h, K, ε)
and a poin an∈Vn∩Gsuch ha |T (an)−w|< δ. In his manne he
sequence {an}nis in G,an→ as n→ ∞ and H{an}n(T, δ, w) is dense in
H(G). The e o e, M(T, {an}n) is dense in H(G) by Rema k 2.1, and he
p oo is inished. ♦
To inish his sec ion we p o ide he nex co olla y (no known up o now)
o Theo em 1.1 and Co olla y 2.8. Φ and Ψ deno e en i e unc ions, wi h Φ o
exponen ial ype i G=Cand o subexponen ial ype i G6=C.
Co olla y 2.10 E e y ope a o T= Φ(D) + Ψ(D−1)which is no iden ically
ze o is omnip esen on H(G).
6
3 Gene al condi ions o DI-ope a o s
As p omised in he i s sec ion, he e we a e going o u nish su icien con-
di ions o an ope a o o be dense-image. Because in he i s h ee esul s
he easoning is e y simila o ha in [4, Sec ion 2], we do no include hei
p oo s. Le us only poin ou ha Lemma 3.1 oge he wi h he Open Mapping
Theo em yields Theo em 3.2.
Lemma 3.1 Le T, S :H(G)→H(G)be ope a o s and A⊂G. Then we
ha e S−1(M(T, A)) = M(TS, A).
Theo em 3.2 Le T, S :H(G)→H(G)be ope a o s, in such a way ha Tis
a DI-ope a o and Sis linea and on o. Then TS is a DI-ope a o .
Co olla y 3.3 I Sis an on o linea ope a o on H(G) hen Sis DI-ope a o .
Be o e p o iding addi ional esul s we conside i con enien o isola e he
ollowing local s abili y condi ion:
Fo e e y K∈ K(G), he e exis s a compac se M⊂G
sa is ying ha o e e y a∈G M, e e y δ > 0 and
e e y ∈H(G) he e exis a closed ball B⊂G K
and α > 0 such ha o e e y g∈H(G)
k −gkB< α =⇒ |T (a)−Tg(a)|< δ.













(P)
Obse e ha , o ins ance, he ope a o s Φ(D) sa is y (P). This is an easy
exe cise i one employs Cauchy’s inequali ies.
Theo em 3.4 Le Tbe an ope a o on H(G)such ha :
(a) M(T, A)6=∅ o all non- ela i ely compac subse s A⊂G.
(b) Tsa is ies condi ion (P).
Then Tis a DI-ope a o .
P oo . Fix a non- ela i ely compac subse A⊂G,w∈C,δ > 0, K∈ K1(G),
h∈H(G) and ε > 0. We ha e o p o e ha HA(T, δ, w)∩D(h, K, ε) is no
emp y. Le Mbe he compac subse associa ed o K u nished by condi ion
(P).
Take a unc ion in M(T, A M) (no e ha A Mis no ela i ely compac
in G). Then ∈HA M(T, δ/2, w), and he e exis s a poin a∈A Msuch
ha
|T (a)−w|<δ
2.(1)
7
By (P), he e exis α > 0 and a closed ball B⊂G Ksuch ha o all
ϕ∈H(G),
kϕ− kB< α =⇒ |Tϕ(a)−T (a)|<δ
2.(2)
Conside he compac se L:= K∪B. No e ha K∩B=∅, so each
connec ed componen o he complemen o Lcon ains a leas one componen
o he complemen o G, as Kdoes. Pick open subse s G1,G2⊂Gwi h
G1∩G2=∅and K⊂G1,B⊂G2. Deno e G0=G1∪G2. The e o e G0is
open and L⊂G0⊂G. De ine he unc ion F:G0→Cas
F(z) = (h(z) i z∈G1
(z) i z∈G2.
Then F∈H(G0), and an applica ion o Runge’s heo em [15, Chap. 13] yields
he exis ence o a a ional unc ion 1wi h poles ou side Gsuch ha
kF− 1kL<min{ε, α}.
Thus, 1∈H(G), and i holds ha
k 1−hkK< ε (3)
and
k 1− kB< α.
The e o e, by (2),
|T 1(a)−T (a)|<δ
2.(4)
Now, by he iangle inequali y, (1) and (4), we ha e
|T 1(a)−w|< δ. (5)
And, joining (3) and (5),
1∈D(h, K, ε)∩HA(T, δ, w).
Consequen ly, he p oo is inished. ♦
When we ha e “wild beha io ” o Ton jus one unc ion and “good beha-
io ” on a dense se o unc ions we a e able o show ha Tis a DI-ope a o .
He e he linea i y is needed.
Lemma 3.5 Le T:H(G)→H(G)be a linea ope a o , ∈∂G and le
{an}nbe a sequence in Gwi h an→ as n→ ∞. Suppose ha he e exis s a
dense subse Do H(G)such ha o each unc ion h∈ D,
∃lim
n→∞(Th)(an)∈C.
Then, i M(T, {an}n)is no emp y, M(T, {an}n)is a dense subse o H(G).
8
P oo . Take ∈M(T, {an}n). We ha e only o p o e ha +h∈M(T, {an}n)
o all h∈ D.
Fix w∈Cand δ > 0. Then he e exis s N∈Nsuch ha
|Th(an)−lh|<δ
2(∀n≥N),(1)
whe e lh= limn→∞ Th(an) (∈C). Since (T )({an}n) = C∞, we also ha e
(T )({an:n≥N}) = C∞so we can ind a na u al numbe m∈Nwi h
m≥Nand
|T (am)−(w−lh)|<δ
2.(2)
The e o e, by (1), (2) and he iangle inequali y, we ha e
|T( +h)(am)−w|=|T (am) + Th(am)−w| ≤
|T (am)−(w−lh)|+|Th(am)−lh|< δ.
So, (T( +h))({an}n) = C∞and his comple es he p oo . ♦
Theo em 3.6 Le Tbe a linea ope a o on H(G)sa is ying
(1) The e exis s a dense subse Din H(G)wi h he p ope y ha o each
h∈ D and each ∈∂G
∃lim
z→ (Th)∈C.
(2) M(T, A)is no emp y o e e y non- ela i ely compac subse A⊂G.
Then Tis a DI-ope a o .
P oo . By Rema k 2.1 we need only p o e ha o each ∈∂G and each
sequence {an}n⊂Gwi h an→ ∞, as n→ ∞, we ha e
(1) + [M(T, {an}n)6=∅] implies [M(T, {an}n) is dense].
Bu his is clea , because (1) gi es us he hypo hesis o Lemma 3.5. So, he
p oo is inished. ♦
Fo ins ance, condi ion (1) in he abo e heo em is sa is ied by a di e-
en ial ope a o Φ(D) and by a ini e-o de an idi e en ial ope a o Ψ(D−1),
whene e Gis a bounded, simply connec ed domain in C: jus ake D=
{polynomials}.
In ou nex s a emen we u nish a su icien condi ion o an ope a o
“poin wise dense ange nea he bounda y” o be dense-image. We mee again
p ope y (P).
9