a Xi :1701.07311 1 [ma h.FA] 25 Jan 2017
SIMULTANEOUS UNIVERSALITY
L. BERNAL-GONZ´
ALEZ AND A. JUNG
Abs ac . In his pape , he no ion o simul aneous uni e sali y is in oduced,
conce ning ope a o s ha ing o bi s ha simul aneously app oxima e any gi en
ec o . This no ion is ela ed o he well known concep s o uni e sali y and
disjoin uni e sali y. Se e al c i e ia a e p o ided, and se e al applica ions o
speci ic ope a o s o sequences o ope a o s a e pe o med, mainly in he se ing
o sequence spaces o spaces o holomo phic unc ions.
1. In oduc ion
In his pape , we a e conce ned wi h he phenomenon o simul aneous app o-
xima ion by he ac ion o se e al ope a o s o , mo e gene ally, by he ac ion o
se e al sequences o mappings. When he exis ence o a dense o bi unde an
ope a o is p o ed, we a e speaking abou uni e sali y o hype cyclici y, see below.
In many si ua ions, i is possible o show he exis ence o one ec o whose o bi s
unde wo o mo e ope a o s app oxima e any gi en ec o . Pushing he ques ion
qui e u he , we wonde unde wha condi ions such app oxima ion akes place
by using a common subsequence. This, oge he wi h i s connec ion wi h o he
kinds o join uni e sali y, will make up he main aim o he p esen manusc ip .
Nex , we ix some ela ed no a ion and e minology o be used in his wo k.
Fo a good accoun o concep s, esul s and his o y conce ning hype cyclici y, he
eade is e e ed o he books [2,21].
By N,N0,R,C,D, B(a, ),B(a, ) (a∈C, > 0) we deno e, espec i ely, he
se o posi i e in ege s, he se N∪ {0}, he eal line, he complex plane, he open
uni disk {z∈C:|z|<1}, he open disk wi h cen e aand adius , and he
co esponding closed disk. Le X, Y be wo Hausdo opological spaces, and
Tn:X→Y(n= 1,2,...) be a sequence o con inuous mappings. Recall ha
(Tn) is said o be uni e sal whene e he e is some (Tn)-o bi which is dense in Y,
ha is, he e exis s an elemen x0∈X–called uni e sal o (Tn)– such ha
{Tnx0:n∈N}=Y.
No e ha Ymus be sepa able. We deno e by U((Tn)) he se o uni e sal ele-
men s o (Tn). When X=Yand T:X→Xis a con inuous sel -mapping,
2010 Ma hema ics Subjec Classi ica ion. 30E10, 47B33, 47A16, 47B38.
Key wo ds and ph ases. hype cyclic ope a o , composi ion ope a o , disjoin uni e sali y, si-
mul aneous uni e sali y.
1
2 BERNAL AND JUNG
hen Tis called uni e sal p o ided ha he sequence (Tn) o i e a es o T(i.e.,
T1=T,T2=T◦T,T3=T◦T2, and so on) is uni e sal, in which case he se
U((Tn)) o uni e sal elemen s will be deno ed by U(T). A sequence Tn:X→Y
(n= 1,2,...) o con inuous mappings is said o be densely uni e sal i U((Tn)) is
dense in X. Bi kho ’s ansi i i y heo em asse s ha , i Xis a Bai e space (in
pa icula , i Xis comple ely me izable) and Yis second-coun able (in pa icula ,
i Xis me izable and sepa able), hen (Tn) is densely uni e sal i and only i (Tn)
is ansi i e ( ha is, gi en nonemp y open se s U⊂X,V⊂Y, he e is N∈N
wi h TN(U)∩V6=∅); i his is he case, hen U((Tn)) is esidual (in ac , a dense
Gδsubse ) in X. I Xlacks isola ed poin s and T:X→Xis uni e sal, hen
U(T) is dense in X(so esidual i Xis, in addi ion, comple ely me izable).
In he case in which Xand Ya e opological ec o spaces o e K(= Ro
C) and (Tn)⊂L(X, Y ) := {linea con inuous mappings X→Y}, he wo ds
hype cyclic and uni e sal a e synonymous, al hough hype cyclic is mos ly used, as
well as he al e na i e no a ion HC((Tn)) := U((Tn)) (and HC(T) := U(T) o
T∈L(X) := L(X, X) = {ope a o s on X}). In pa icula , we ha e i Xand Y
a e F-spaces wi h Ysepa able, hen HC((Tn)) (HC(T), wi h Xsepa able, esp.)
is esidual in Xas soon as (Tn) is ansi i e (as soon as Tis hype cyclic, esp.).
Recall ha an F-space is a comple ely me izable opological ec o space.
Assume now ha X, Y a e opological spaces, wi h Xa Bai e space and Y
second-coun able, and ha Sn:X→Yand Tn:X→Y(n∈N) a e densely
uni e sal sequences. Since U((Sn)),U((Tn)) a e dense Gδsubse s o X, we ha e
ha U((Sn)) ∩ U((Tn)) is also dense, so non-emp y. Hence he e is a common
hype cyclic elemen x∈X. So, o a gi en poin y∈Y, he e a e sequences
{n1< n2<· · ·} and {m1< m2<· · · } in Nsuch ha
Snjx→yand Tmjx→yas j→ ∞.
Then he ollowing ques ion a ises na u ally:
Unde wha condi ions on (Sn)and (Tn)one can gua an ee he exis-
ence o an elemen x∈Xsuch ha , o any gi en y∈Y, he e is
one sequence {n1< n2<· · · } ⊂ Nsuch ha
Snjx−→ y←− Tnjxas j→ ∞?
O cou se, a simila ques ion can be posed o ini ely many sequences and o
ini ely many single ope a o s on X, jus by conside ing he sequences o hei
i e a es in he la e case. Wi h his in mind, he new concep o simul aneous
uni e sali y will be in oduced in he nex sec ion, and compa ed o o he ela ed
no ions exis ing in he li e a u e, such as hose o disjoin hype cyclici y and he
weakly mixing p ope y. Se e al su icien condi ions o simul aneous uni e sal-
i y/hype cyclici y will be p o ided in Sec ion 3. Examples o ini e amilies o
simul aneous hype cyclic ope a o s will be u nished in sec ions 4–6, s a ing wi h
SIMULTANEOUS UNIVERSALITY 3
mul iples o an ope a o and ending up in he amewo ks o sequence spaces and
o spaces o analy ic unc ions on complex domains.
2. Simul aneously uni e sal sequences
Le us de ine he new concep ha is he ma e o his pape . I p∈Nand
Yis a nonemp y se , hen by ∆(Yp) we deno e he diagonal o Yp=Y× · · · × Y
(p imes), ha is, he subse ∆(Yp) = {(y,y,...,y) : y∈Y}. I Yis a opological
space, hen Ypis assumed o be endowed wi h he p oduc opology.
De ini ion 2.1. Le p∈Nand X, Y be Hausdo opological spaces. Assume
ha , o each j∈ {1,...,p},Tj,n :X→Y(n∈N) is a sequence o con inuous
mappings. Conside he sequence
[T1,n,...,Tp,n] : x∈X7−→ (T1,nx,...,Tp,nx)∈Yp(n∈N).
Le also T1,...,Tp:X→Xbe con inuous mappings.
(a) We say ha he sequences (T1,n),...,(Tp,n) a e simul aneously uni e sal
(o s-uni e sal ) whene e he e exis s an elemen x0∈X–called s-uni e sal
o (T1,n),...,(Tp,n)– sa is ying
{[T1,n,...,Tp,n]x0:n∈N} ⊃ ∆(Yp).
The se o such s-uni e sal elemen s will be deno ed by s-U((T1,n),...,(Tp,n)).
(b) The sequences (T1,n),...,(Tp,n) a e said o be densely simul aneously uni-
e sal i he se s-U((T1,n),...,(Tp,n)) is dense in X. And hey a e called
he edi a ily simul aneously uni e sal (he edi a ily densely simul aneously
uni e sal, esp.) i , o e e y s ic ly inc easing sequence (nk)⊂N, he
sequences (T1,nk), . . ., (Tp,nk) a e s-uni e sal (densely s-uni e sal, esp.).
(c) The mappings T1, . . . , Tpa e called s-uni e sal (densely s-uni e sal,he e-
di a ily s-uni e sal,he edi a ily densely s-uni e sal, esp.) i he sequences
(Tn
1),...,(Tn
p) a e s-uni e sal (densely s-uni e sal, he edi a ily s-uni e sal,
he edi a ily densely s-uni e sal, esp.). The se s-U((Tn
1),...,(Tn
p)) o co -
esponding s-uni e sal elemen s will be deno ed by s-U(T1,...,Tp).
Rema ks 2.2. 1. I Yis i s -coun able (in pa icula , i Yis me izable), hen
he s-simul aneous uni e sali y o (Tj,n)n∈N(1 ≤j≤p) means he exis ence o
some x0∈Xenjoying he p ope y ha , o e e y y∈Y, he e is a (s ic ly
inc easing) sequence (nk)⊂Nsuch ha Tj,nkx0→yas k→ ∞ (j= 1,...,p).
2. In [18, Kapi el 1] he no ion o ela i e uni e sali y on a closed subse o he
a i al space is in oduced unde e y gene al assump ions. In he p esen pape
we s udy a special case o his si ua ion (no e ha ∆(Yp) is closed in Ypsince Yp
is Hausdo ) unde mo e speci ic hypo heses.
3. Acco ding o he in oduc ion, i X, Y a e opological ec o spaces and Tj,n, Tj∈
L(X, Y ) (j= 1,...,p;n∈N), hen we use he exp essions “s-hype cyclic”,
4 BERNAL AND JUNG
“densely s-hype cyclic” and “he edi a ily densely s-hype cyclic” a he han “s-
uni e sal”, “densely s-uni e sal” and “he edi a ily densely s-uni e sal”, espec-
i ely. In addi ion, we will deno e s-HC((T1,n),...,(Tp,n)) :=
s-U((T1,n),...,(Tp,n)) and s-HC(T1,...,Tp) := s-U(T1,...,Tp) in his case.
4. Fo a single ope a o T, hype cyclici y (he edi a y hype cyclici y, esp.) is
equi alen o dense hype cyclici y (he edi a y dense hype cyclici y, esp.).
5. The p ope y o simul aneous uni e sali y o (T1,n),...,(Tp,n) is weake han
he p ope y ha he sequence ([T1,n,...,Tp,n]) is subspace-uni e sal o ∆(Yp),
meaning ha he se {[T1,n,...,Tp,n]x0:n∈N} ∩ ∆(Yp) is dense in ∆(Yp) o
some x0∈X(see e.g. [1,22,24] o esul s on subspace-hype cyclici y/uni e sali y).
Be o e going on, we wan o compa e s-uni e sali y o o he ela ed concep s
de ined in he li e a u e. In 2007, B`es, Pe is and he i s au ho ([11],[4]) in o-
duced he no ion o disjoin (o d-) uni e sali y (some imes called d-hype cyclici y
in he men ioned e e ences). Unde he same assump ions and e minology as in
De ini ion 2.1, he sequences (T1,n),...,(Tp,n) a e said o be d-uni e sal whene e
he sequence [T1,n . . . , Tp,n] : X→Yp(n∈N) is uni e sal, ha is, whene e he e
exis s some x0∈Xsuch ha he join o bi {(T1,nx0,...,Tp,nx0) : n∈N}is
dense in Yp. As a ma e o ac , d-uni e sali y should no be con used wi h he
uni e sali y o he sequence
T1,n ⊕ · · · ⊕ Tp,n : (x1,...,xp)∈Xp7−→ (T1,nx1,...,Tp,nxp)∈Yp.
T i ially, disjoin uni e sali y o (T1,n),...,(Tp,n) implies uni e sali y o he las
sequence as well as simul aneous uni e sali y o (T1,n),...,(Tp,n). Also, i ially,
s-uni e sali y implies he uni e sali y o each sequence (Tj,n)n∈N(j= 1,...,p)
(in pa icula , Ymus be sepa able). Bu no o he implica ions among hese
p ope ies hold, e en conside ing only p= 2 and sequences o i e a es o single
ope a o s. The ollowing examples illus a e his si ua ion:
1. Assume ha Tis a hype cyclic ope a o on a opological ec o space.
Then he ope a o s T, T a e s-hype cyclic bu no d-hype cyclic.
2. In 1969, S. Rolewicz [26] p o ed ha i c∈Khas modulus >1 and B
is he backwa d shi (xn)∈ℓ27→ (xn+1)∈ℓ2, hen he ope a o cB is
hype cyclic. In pa icula , he ope a o s T= 2Band S= 4B= 2Ta e
hype cyclic, bu T, S a e clea ly no s-hype cyclic.
3. Since each o he ope a o s T, S o he la e example is mixing (see he
de ini ion a he beginning o he nex sec ion, ega ding he sequences o
i e a es; see also [21, p. 46]), he ope a o T⊕Sis hype cyclic, bu T, S
a e no s-hype cyclic.
4. De la Rosa and Read [15] we e able o cons uc a Banach space Xand an
ope a o T∈L(X) such ha Tis hype cyclic (hence T, T a e s-hype cyclic)
bu Tis no weakly mixing on X, meaning ha T⊕Tis no hype cyclic
on X2.
SIMULTANEOUS UNIVERSALITY 5
While d-hype cyclic ope a o s mus be subs an ially di e en , s-hype cyclici y
allows mo e simila i y. Fo ins ance, an ope a o can ne e be d-hype cyclic wi h a
scala mul iple o i sel (see [11, p. 299]). Ne e heless, s-hype cyclici y is possible
in conc e e si ua ions. This will be analyzed in Sec ion 4. Sec ions 5 and 6 a e
de o ed o mo e speci ic ope a o s, namely backwa d shi s and ope a o s on spaces
o analy ic unc ions.
We close his sec ion by es ablishing, unde app op ia e assump ions, he exis-
ence o la ge ec o subspaces consis ing, excep o ze o, o s-hype cyclic ec o s.
Theo em 2.3. (a) Le Xbe a opological ec o space and Tj∈L(X) (j=
1,...,p). I T1, . . ., Tpa e s-hype cyclic and a leas one o hem commu es
wi h he o he s, hen s-HC(T1,...,Tp)con ains, excep o 0, a dense lin-
ea subspace o X.
(b) Le Xand Ybe wo opological ec o spaces such ha Yis me izable.
Assume ha (Tj,n)⊂L(X, Y ) (j= 1,...,p)a e he edi a ily s-hype cyclic
sequences. Then s-HC((T1,n),...,(Tp,n)) con ains, excep o 0, an in ini e
dimensional ec o subspace o X.
(c) Le Xand Ybe wo me izable sepa able opological ec o spaces. Assume
ha (Tj,n)⊂L(X, Y ) (j= 1,...,p)a e he edi a ily densely s-hype cyclic
sequences. Then s-HC((T1,n),...,(Tp,n)) con ains, excep o 0, a dense
linea subspace o X.
P oo . (a) By hypo hesis, he e is i∈ {1,...,p}such ha TiTj=TjTi(j=
1,...,p). The e o e P(Ti)Tj=TjP(Ti) o all jand e e y polynomial Pwi h
coe icien s in K. Le Pdeno e he se o such polynomials. O cou se, he
ope a o Tiis hype cyclic. F om a esul by Wengen o h [29], he ope a o P(Ti)
has dense ange as soon as P∈ P {0}. Pick any x0∈s-HC(T1,...,Tp). Le
us de ine M:= {P(Ti)x0:P∈ P {0}}. Then Mis a linea subspace o X.
I is dense because Mcon ains he o bi {Tn
ix0:n∈N}, ha is dense in Xas
x0∈HC(Ti). I emains o show ha M {0} ⊂ s-HC(T1,...,Tp).
To his end, ix u∈M {0}. Then he e is P∈ P {0}such ha u=P(Ti)x0.
I mus be p o ed ha
Z⊃∆(Xp),
whe e Z:= {(Tn
1u, . . . , Tn
pu) : n∈N}={(P(Ti)Tn
1x0,...,P(Ti)Tn
px0) : n∈N},
whe e he las equali y ollows om commu a i i y. We know ha ∆(Xp)⊂
{(Tn
1x0,...,Tn
px0) : n∈N}. Le A:= {(Tn
1x0,...,Tn
px0) : n∈N},ϕ:= P(Ti)
and Φ : Xp→Xpbe he mapping de ined as Φ(x1,...,xp) := (ϕ(x1),...,ϕ(xp)).
Then, as ϕis con inuous, we ge
Z= Φ(A)⊃Φ(A)⊃Φ(∆(Xp)) = {(ϕ(x),...,ϕ(x)) : x∈X},
so Z⊃{(ϕ(x),...,ϕ(x)) : x∈X}. Gi en y∈Xand a neighbo hood Uo
(y,y,...,y), he e exis s a neighbo hood Vo ysuch ha U⊃Vp. Since ϕ

6 BERNAL AND JUNG
has dense ange, one can ind x∈Xwi h ϕ(x)∈V. Then (ϕ(x),...,ϕ(x)) ∈
U. In o he wo ds, (y,...,y)∈{(ϕ(x),...,ϕ(x)) : x∈X}, so (y,...,y)∈Z.
Consequen ly, Z⊃∆(Xp), as equi ed.
(b)–(c). By mimicking he p oo s o Theo ems 1–2 o [3] (in which he esul s
a e gi en o a single sequence (Tn)), we can cons uc ecu si ely a sequence
(xN)N∈N⊂Xand a amily {(q(N, k))k∈N:N∈N0}o s ic ly inc easing
subsequences o Nsa is ying, o all N∈N, he ollowing condi ions: xN∈
GN∩s-HC((T1,q(N−1,k),...,(Tp,q(N−1,k))) and Tj,q(l,k)xN→0 as k→ ∞ o all
l≥Nand all j∈ {1,...,p}, whe e G0:= Xand GN:= X span {x1,...,xN−1}
(N∈N) i he assump ions o (b) hold, while {GN}N∈Ndeno es any ixed open
basis o Xi he assump ions o (c) hold. Then M:= span {xN:N∈N}is he
sough -a e ec o subspace. The de ails a e le as an exe cise. 
3. s-Uni e sali y c i e ia
A numbe o wo kable su icien condi ions will be use ul o de ec s-uni e sali y.
Recall ha a sequence o con inuous mappings Tn:X→Y(n∈N) is called mix-
ing p o ided ha ,gi en nonemp y open se s U⊂X,V⊂Y, he e is N∈Nsuch
ha Tn(U)∩V6=∅ o all n≥N. The co esponding no ion o simul aneous
mixing p ope y a ises na u ally, as well as he one o simul aneous ansi i i y.
No e ha Tn(U)∩V6=∅is equi alen o U∩T−1
n(V)6=∅.
De ini ion 3.1. Le p∈Nand X, Y be Hausdo opological spaces. Assume
ha , o each j∈ {1,...,p},Tj,n :X→Y(n∈N) is a sequence o con inuous
mappings. Le also T1,...,Tp:X→Xbe con inuous mappings. We say ha :
(a) The sequences (T1,n),...,(Tp,n) a e simul aneously ansi i e (o s- ansi i e)
p o ided ha , o e e y pai o nonemp y open se s U⊂X,V⊂Y, he e
is N∈Nsuch ha U∩Tp
j=1 T−1
j,N (V)6=∅.
(b) The sequences (T1,n),...,(Tp,n) a e simul aneously mixing (o s-mixing)
p o ided ha , o e e y pai o nonemp y open se s U⊂X,V⊂Y, he e
is N∈Nsuch ha U∩Tp
j=1 T−1
j,n (V)6=∅ o all n≥N.
(c) The mappings T1,...,Tpa e simul aneously ansi i e (simul aneously mix-
ing, esp.) whene e he sequences (Tn
1),...,(Tn
p) a e s- ansi i e (s-mixing,
esp.).
Rema k 3.2. Co esponding concep s o d- ansi i i y and d-mixing we e in o-
duced in [11], whe e Tp
j=1 T−1
j,n (Vj) (Vjnonemp y open subse s o Y,j= 1,...,p)
appea s ins ead o Tp
j=1 T−1
j,n (V). Also, mos c i e ia gi en in his sec ion ha e hei
coun e pa s o he ela ed d-p ope ies as p o ided in [4] and [11]. A ho ough
s udy o d-mixing ope a o s is p o ided in [8].
No e ha , con a y o he one-sequence case, he ac s U∩Tp
j=1 T−1
j,N (V)6=∅
and Tp
j=1 Tj,N (U)∩V6=∅a e no equi alen . Obse e also ha Tp
j=1 T−1
j,n (V) =
SIMULTANEOUS UNIVERSALITY 7
[T1,n,...,Tp,n]−1(Vp). F om he de ini ions, i is easy o check ha he sequences
(T1,n),...,(Tp,n) a e s-mixing i and only i , o e e y s ic ly inc easing sequence
(nk) in N, he sequences (T1,nk),...,(Tp,nk) a e s- ansi i e. The ollowing p opo-
si ion p o ides wha can be called he Bi kho s- ansi i i y heo em.
P oposi ion 3.3. Unde he same assump ions and e minology as in De ini ion
3.1, le us suppose, in addi ion, ha Xis Bai e and Yis second-coun able. Then
we ha e:
(i) The sequences (T1,n),...,(Tp,n)a e s- ansi i e i and only i hey a e densely
s-uni e sal. I his is he case, hen he se s-U((T1,n),...,(Tp,n)) is esidual
in X.
(ii) The sequences (T1,n),...,(Tp,n)a e s-mixing i and only i , o e e y s ic ly
inc easing sequence (nk)⊂N, he sequences (T1,nk),...,(Tp,nk)a e densely
s-uni e sal.
P oo . Pa (ii) is an immedia e consequence o (i). Le us p o e (i). Fix a
coun able open basis (Vm) o Y, as well as a poin x0∈X. Then x0∈s-
U((T1,n),...,(Tp,n)) i and only i , gi en a nonemp y open se V⊂Y, he e is
n∈Nwi h [T1,n,...,Tp,n]x0∈Vp, ha is, x0∈Sn∈NTp
j=1 T−1
j,n (V). Since each V
con ains some Vmand each Vmis a nonemp y subse o Y, he las p ope y is he
same as x0∈Tm∈NSn∈NTp
j=1 T−1
j,n (Vm), which shows ha
s-U((T1,n),...,(Tp,n)) =
m∈N[
n∈N
p
j=1
T−1
j,n (Vm).(1)
Since he Tj,n’s a e con inuous, each se Tp
j=1 T−1
j,n (Vm) is open. I (T1,n),...,(Tp,n)
a e s- ansi i e hen e e y se Sn∈NTp
j=1 T−1
j,n (Vm) (m∈N) is (open and) dense.
Hence hei (coun able) in e sec ion, which equals s-U((T1,n),...,(Tp,n)) by (1), is
a dense Gδsubse (so esidual) in Xbecause Xis Bai e. Con e sely, assume ha
he se o s-uni e sal elemen s is dense in Xand ix a nonemp y open subse Vo
Y. Then he e is m∈Nwi h V⊃Vm. I ollows om (1) ha Sn∈NTp
j=1 T−1
j,n (Vm)
is dense in X, so he bigge se Sn∈NTp
j=1 T−1
j,n (V) is also dense. Bu his means
ha , gi en a nonemp y se U⊂X, he e is N∈Nsuch ha U∩Tp
j=1 T−1
j,N (V)6=∅
o , in o he wo ds, he sequences (T1,n),...,(Tp,n) a e s- ansi i e. 
In he linea case, we s a e he ollowing se o su icien condi ions, ha a e
inspi ed by he esul s con ained in [19, Sec . 1c] and he e e ences ci ed in i .
Theo em 3.4. Le Xand Ybe opological ec o spaces such ha Xis Bai e and
Yis me izable and sepa able, and le (Tj,n)n∈N(j= 1,...,p)be sequences in
L(X, Y ). Assume ha he e a e espec i e dense subse s X0o Xand Y0o Y
sa is ying a leas one o he ollowing condi ions:
8 BERNAL AND JUNG
(A) Fo e e y pai o ec o s x∈X0, y ∈Y0, he e exis sequences (nk)⊂N
and (xk)⊂Xwi h xk→0,Tj,nkx→0and Tj,nkxk→y(j= 1,...,p)
as k→ ∞.
(B) Fo e e y x∈X0, he sequences (Tj,nx)n∈N(j= 1,...,p)con e ge in Y
o a common limi and, o e e y y∈Y0, he e exis sequences (nk)⊂N
and (xk)⊂Xwi h xk→0and Tj,nkxk→y(j= 1,...,p)as k→ ∞.
(C) Fo e e y x∈X0, he e exis s a sequence (nk)⊂Nsuch ha he sequences
(Tj,nkx)k∈N(j= 1,...,p)con e ge in Y o a common limi and, o e e y
y∈Y0, he e exis s a sequence (xn)⊂Xsuch ha xn→0and Tj,nxn→y
(j= 1,...,p)as n→ ∞.
Then (Tj,n)n∈N(j= 1,...,p)a e densely s-hype cyclic.
P oo . Acco ding o P oposi ion 3.3, we should show ha (Tj,n)n∈N(j= 1,...,p)
a e s- ansi i e. Wi h his aim, ix a pai o nonemp y open se s U⊂X,V⊂Y.
We should exhibi an N∈Nsuch ha U∩Tp
j=1 T−1
j,N (V)6=∅.
Assume i s ha (A) holds. By densi y, he e a e x∈X0and y∈Y0such
ha x∈Uand y∈V. De ine A:= U−xand B:= V−y. Then Aand B
a e open neighbo hoods o 0 in Xand Y espec i ely. Take a 0-neighbo hood
C⊂Ysa is ying C+C⊂B. Conside he sequences (nk) and (xk) p o ided
by (A). Then he e is k∈Nsuch ha xk∈A,Tj,nkx∈Cand Tj,nkxk∈y+C
(j= 1,...,p). Le u:= x+xkand N:= nk. We ge u∈x+A=Uand
Tj,N u=Tj,N x+Tj,N xk∈C+y+C⊂y+B=V(j= 1,...,p), so ha
u∈U∩Tp
j=1 T−1
j,N (V).
Suppose now ha (B) holds. By densi y, he e is x∈X0such ha x∈U.
De ine A:= U−x, a neighbo hood o 0. By hypo hesis, he e is z∈Ysuch ha
Tj,n →zas n→ ∞ (j= 1,...,p). Since Y0is dense in Y, he e is y∈Y0wi h
y∈z+V. Le B:= V−y+z, a neighbo hood o 0 in Y. Take a 0-neighbo hood
C⊂Ysa is ying C+C⊂B. We ha e ha Tj,n ∈z+C(j= 1,...,p) o n≥n0,
say. Conside he sequences (nk) and (xk) p o ided by (B) o he ec o y−z, so
ha xk→0 and Tj,nkxk→y−z(j= 1,...,p) as k→ ∞. Choose k∈Nso la ge
ha nk≥n0,xk∈Aand Tj,nkxk∈y−z+C(j= 1,...,p). Le u:= x+xkand
N:= nk. Then u∈x+A=Uand, o e e y j= 1,...,p,
Tj,N u=Tj,N x+Tj,N xk∈z+C+y−z+C=y+C+C⊂y+B=V,
so ha u∈U∩Tp
j=1 T−1
j,N (V), as equi ed. Unde assump ion (C), he p oo is
simila and le as an exe cise. 
Two o he mos popula c i e ia o hype cyclici y a e he so-called blow-up/col-
lapse c i e ion and he hype cyclici y c i e ion (see [2,20,21]). Now, we can ob ain
hei espec i e s- e sions.
P oposi ion 3.5. [s-Blow-up/Collapse C i e ion] Le Xbe a Bai e me izable
sepa able opological ec o space, and le (Tj,n)n∈N(j= 1,...,p)be sequences
SIMULTANEOUS UNIVERSALITY 9
in L(X). Suppose ha , o e e y nonemp y open subse s U, V o Xand e e y
0-neighbo hood W⊂X he e is N∈Nsuch ha
W∩
p
j=1
T−1
j,N (V)6=∅6=U∩
p
j=1
T−1
j,N (W).
Then (Tj,n)n∈N(j= 1, . . . , p)a e densely s-hype cyclic.
P oo . Fix a pai o nonemp y open se s U, V ⊂X. Choose ec o s x∈U,y∈V.
I su ices o exhibi sequences sequences (nk)⊂Nand (xk)⊂Xwi h xk→x
and Tj,nkxk→y(j= 1, . . . , p), because his would en ail he exis ence o some
k∈Nsuch ha xk∈Uand Tj,nkxk∈V(j= 1,...,p), so xk∈U∩Tp
j=1 T−1
j,nk(V).
In o he wo ds, he sequences (Tj,n)n∈N(j= 1,...,p) would be s- ansi i e, hence
densely s-hype cyclic by P oposi ion 3.3.
Wi h his aim, choose a undamen al dec easing sequence (Wk) o 0-neighbo -
hoods. Then (Uk) := (x+Wk) and (Vk) := (y+Wk) a e undamen al dec easing
sequences o x-neighbo hoods and y-neighbo hoods, espec i ely. By hypo hesis,
o each k∈N, he e a e nk∈Nand poin s x′
kand x′′
ksuch ha x′
k∈Wk∩
Tp
j=1 T−1
j,nk(Vk) and x′′
k∈Uk∩Tp
j=1 T−1
j,nk(Wk). Le xk:= x′
k+x′′
k. Then xk→x
as k→ ∞ because x′
k∈Wk(so x′
k→0) and x′′
k∈Uk(so x′′
k→x). Finally,
Tj,nkxk=Tj,nkx′
k+Tj,nkx′′
k→y+ 0 = y(j= 1,...,p) because Tj,nkx′
k∈Vkand
Tj,nkx′′
k∈Wk o all k∈N.
Recall ha he con ex hull con (A) o a subse Ao a ec o space Xis he
leas con ex subse o Xcon aining A.
De ini ion 3.6. Le Xbe a Bai e me izable sepa able locally con ex space,
(nk)⊂Nbe a s ic ly inc easing sequence and Tj∈L(X) (j= 1,...,p). We
say ha T1,...,Tpsa is y he s-hype cyclici y c i e ion wi h espec o (nk) i
he e a e subse s X0⊂X, W0⊂Xpsuch ha X0is dense in Xand
W0⊃∆(Xp)
as well as mappings Rk:W0→X(k∈N) such ha
(i) Tnk
j→0 poin wise on X0as k→ ∞ (j= 1,...,p),
(ii) Rk→0 poin wise on W0as k→ ∞ and
(iii) Fo e e y w= (w1,...,wp)∈W0and e e y j∈ {1,...,p} he e is yj∈
con ({w1,...,wp}) such ha Tnk
jRkw→yjas k→ ∞.
Theo em 3.7. [s-Hype cyclici y C i e ion] Le Xbe a Bai e me izable sepa a-
ble locally con ex space and Tj∈L(X) (j= 1,...,p). I T1,...,Tpsa is y he
s-hype cyclici y c i e ion wi h espec o some (nk)⊂N, hen (Tnk
1),...,(Tnk
p)a e
s-mixing. In pa icula , T1,...,Tpa e densely s-hype cyclic.
P oo . Le U, V ⊂Xbe nonemp y open se s. Then he e a e x0∈U∩X0and
y0∈V. By local con exi y, he e is a con ex open se e
Vwi h y0∈e
V⊂V. As
16 BERNAL AND JUNG
(T2k+1) and ((−T)2k+1) = (−T2k+1) is clea ly no possible. In connec ion wi h
his, i is s a ed in [7, Rema k 24(ii)] and ac ually p o ed in [28, P oposi ion
4.9] ha in case o unimodula scala s c1,...,cpe e y d-hype cyclic ec o x0 o
T1,...,Tpis also d-hype cyclic o c1T1,...,cpTp. The p oo uses c ucially he ac
ha such a ec o x0sa is ies (x0,...,x0)∈HC(T1⊕ · · · ⊕ Tp). Thus, i canno
be adap ed o s-hype cyclici y. Hence, we pose he ques ion: Does he equali y
s-HC(T1,...,Tp) = s-HC(c1T1,...,cpTp) hold?
3. Conce ning again pa (b) and ega ding i s p oo , we may ob ain a much
s onge esul in he case K=Cand Xa Banach space. Recall ha a
nonemp y subse E⊂Cis said o be pe ec i i is closed and each poin
o Eis an accumula ion poin o E. In pa icula , e e y pe ec se is un-
coun able. I is well known (see [13, Theo em 8.138(b)]) ha he e a e pe ec
Di ichle subse s o T. We ha e ha i E⊂Tis a pe ec Di ichle se and
T∈L(X) is mixing, hen he uncoun able amily o o a ions {cT :c∈E∪ {1}}
is densely uni o mly s-hype cyclic, in he sense ha he e is a dense se o ec o s
x0∈Xsa is ying he ollowing: o e e y y∈X he e is (nk)⊂Nsuch ha
limk→∞ supc∈E∪{1}k(cT)nkx0−yk= 0. Indeed, we can ake a sequence (mk)⊂N
such ha supc∈E∪{1}|cmk−1|= supc∈E|cmk−1| → 0 as k→ ∞. As Tis
mixing, he se HC((Tmk)) is dense. I x0∈HC((Tmk)), hen he e is a subse-
quence (nk)⊂(mk) wi h Tmkx0→y. The conclusion ollows om he inequali y
k(cT)nkx0−yk ≤ kcnk(Tnkx0−y)k+k(cnk−1)yk.
4. P oposi ion 4.1 u nishes examples o pai s o ope a o s –on spaces o sequences
o o holomo phic unc ions (see sec ions 5–6)– ha a e s-hype cyclic bu no d-
hype cyclic: he mul iples 2B, −2Bo he backwa d shi Bon ℓq(1 ≤q < ∞) o
c0;D, −Don H(C) (D := ′); Cϕ,−Cϕon H(G), whe e Cϕ := ◦ϕ,G⊂Cis
a simply connec ed domain and ϕis a un-away au omo phism o G.
5. Backwa d shi s and s-hype cyclici y
In his sec ion, we conside he sequence spaces c0and ℓq(1 ≤q < ∞) o e
K=Ro C. I a= (an)n∈Nis a bounded sequence in K {0}, hen Bawill deno e
he weigh ed backwa d shi
Ba: (x0, x1, x2,...)∈X7→ (a1x1, a2x2,...)∈X
on X=c0o ℓq. The unweigh ed backwa d shi Bis B=Ba, whe e a=
(1,1,1,...). Salas cha ac e ized he hype cyclici y o Bain e ms o he weigh
sequence a. B`es and Pe is [11, Theo em 4.1] did he same o he d-hype cyclici y
o di e en powe s o Ba. This cha ac e iza ion happens o hold also o s-
hype cyclici y.
P oposi ion 5.1. Le X=c0o ℓq(1 ≤q < ∞),p≥2and le 1,..., p∈N
wi h 1< 2<···< pbe gi en. Fo each l∈ {1,...,p}, le al= (al,n)n∈Nbe a
weigh sequence. Then he ollowing a e equi alen :

SIMULTANEOUS UNIVERSALITY 17
(i) B 1
a1,...,B p
apa e d-hype cyclic.
(ii) B 1
a1,...,B p
apa e s-hype cyclic.
(iii) Fo e e y M > 0and e e y k∈N he e is m∈Nsa is ying, o
each j∈ {0,1, . . . , k}, ha |al,j+1 ···al,j+ lm|> M (1 ≤l≤p)and
|al,j+1 ··· al,j+ lm|
|as,j+( l− s)m+1 ··· as,j+ lm|> M (1 ≤s < l ≤p).
(i ) B 1
a1,...,B p
apsa is y he d-hype cyclici y c i e ion.
( ) B 1
a1,...,B p
apsa is y he s-hype cyclici y c i e ion.
P oo . The equi alence o (i), (iii) and (i ) is p o ed in [11, Theo em 4.1]. Tha
(i) implies (ii) is i ial. Mo eo e , (ii) ⇒(iii) is p o ed in ac in he p oo o “(a)
⇒(b)” o he same e e ence, since only he simul aneous app oxima ion o one
ec o (namely e0+···+eq) is used. Finally, we clea ly ha e (i ) ⇒( ) ⇒(ii). 
Rema ks 5.2. 1. An analogous esul abou equi alence o d- and s-hype cyclici y
also wo ks o powe s o weigh ed bila e al shi s (see Theo em 4.7 o [11] and i s
p oo ).
2. Co olla y 4.4 in [11] also wo ks wi h jus s-uni e sali y, as i is a consequence
o Theo em 4.1 he e. In pa icula , we ha e ha Ba, B2
a,...,Bp
aa e s-hype cyclic
on Xi and only i Ba⊕B2
a⊕· · ·⊕Bp
ais hype cyclic on Xp. B`es, Ma in and Pe is
[7, p. 855] cons uc ed an ope a o T:= Baon ℓ2such ha Tis hype cyclic bu
T⊕T2is no hype cyclic on ℓ2⊕ℓ2, so ha T, T2is no d-hype cyclic on ℓ2. Then
we ob ain ha T, T2a e e en no s-hype cyclic. Acco ding o [21, Theo em 4.8],
he men ioned T=Bais no mixing. In [8, Sec . 3], a mixing ope a o T∈L(ℓ2)
o which T, T2a e no d-mixing is exhibi ed. Bu he exis ence o a mixing Ton
a sepa able Banach space such ha T, T 2a e no d-hype cyclic is unknown so a
[8, Ques ion 3.7].
A mo e delica e ques ion a ises when 1≤ 2≤ · · · ≤ p. In [11, Co ol-
la y 4.2], he ollowing is p o ed o weigh ed powe s o he unweigh ed backwa d
shi : i p≥2 and l∈N,λl∈K(1 ≤l≤p) wi h 1≤ 2≤ · · · ≤ p,
hen λ1B 1,...,λpB pa e d-hype cyclic i and only i 1< 2<··· < pand
1<|λ1|<|λ2|<··· <|λp|. The ollowing esul shows ha s-hype cyclici y is
possible unde sligh ly weake assump ions.
P oposi ion 5.3. Le p≥2, and le l∈N,λl∈K(1 ≤l≤p)wi h 1≤ 2≤
· · · ≤ p. Le Adeno e he se A:= {j∈ {1,...,p−1}: j= j+1}and conside
he condi ions
(i) 1 <|λj| o all j∈ {1,...,p},
(ii) |λj|<|λj+1| o all j∈ {1,...,p−1} A,
(iii) |λj|=|λj+1| o all j∈A.
Then λ1B 1,...,λpB pa e s-hype cyclic on X=c0o ℓq(1 ≤q < ∞)i and only
i (i),(ii) and (iii) hold.
18 BERNAL AND JUNG
P oo . Fi s , suppose ha condi ions (i),(ii) and (iii) hold. We w i e {1,...,p} A=
{ 1,..., d}, wi h d∈Nand 1<··· < d. As he se {λi/λj:i, j ∈ {1,...,p}
wi h |λi|=|λj|} ⊂ Tis ini e, i is a Di ichle se . Hence he e exis s a s ic ly
inc easing sequence (nk)⊂Nsuch ha
λi
λjnk
→1 (k→ ∞) o all i, j ∈ {1,...,p}wi h |λi|=|λj|.(1)
Conside he se X0o ini e sequences, ha is, X0:= c00 ={x= (xn)∈X:
exis s n0=n0(x)∈Nsuch ha xn= 0 o all n≥n0}. Then X0is dense in
X. I we se W0:= ∆(Xp
0)⊂Xp, hen W0= ∆(Xp
0)⊃∆(Xp) because X0is
dense in X. Now, we se Tj:= λjB j(j= 1,...,p). De ine, o each k∈N, he
mapping Rk:W0→Xas ollows. I x= (x1, x2,...,xN,0,0,0,...)∈X0and
w= (x,x,...,x), hen
Rkw=(01, u1,02, u2,...,0N, uN,0,0,0,...) i nk≥N
(0,0,0,...) i nk< N,
whe e 01:= (0,0, . . . , 0) [ 1nk imes], 0l:= (0,0,...,0) [( l− l−1)nk−N imes]
i l≥2 and ul:= 1
λnk
l
x1,..., 1
λnk
l
xN(l≥1). We ha e:
(a) Fo each j∈ {1,...,p}and each x= (x1, x2,...,xN,0,0,0,...)∈X0,
Tnk
jx= 0 as soon as jnk> N, so Tnk
j→0 (k→ ∞) poin wise on X0.
(b) Fo e e y w= (x,...,x)∈W0as be o e, he de ini ion o Rk oge he
wi h (i) yields Rkw→0 as k→ ∞.
(c) Fix w= (x,...,x)∈W0, whe e x= (x1, x2,...,xN,0,0,0,...). Fo e e y
j∈ {1,...,p} he e is exac ly one l∈ {1,...,d}such ha |λj|=|λ l|, due
o (ii) and (iii). Finally, i nk≥N, we ha e
Tnk
jRkw=λj
λ lnkx1,λj
λ lnkx2,...,λj
λ lnkxN,0,0,...,0,
λj
λ l+1 nkx1,λj
λ l+1 nkx2,...,λj
λ l+1 nkxN,0,0,...,0,...,
λj
λ dnkx1,λj
λ dnkx2,...,λj
λ dnkxN,0,0,0,0,....
I ollows om (ii) ha ( λj
λ s)nkxν→0 as k→ ∞ o all s∈ {l+1,...,d}and
all ν∈ {1, . . . , N}, while (1) en ails ha ( λj
λ l
)nkxν→xνas k→ ∞ o all
ν∈ {1,...,N}. Consequen ly, Tnk
jRkw→(x1, x2,...,xN,0,0,0,...) = x.
An applica ion o he s-hype cyclici y c i e ion (see also Rema k 3.8.1) concludes
he i s pa o he p oo .
Now, suppose ha λ1B 1,...,λpB pa e s-hype cyclic. Since hype cyclic ope a-
o s on no med spaces ha e no m la ge han 1, we ob ain
1<kλjB jk=|λj|kB jk=|λj|
SIMULTANEOUS UNIVERSALITY 19
o all j= 1,...,p (c . he p oo o Co olla y 4.2 in [11]), i.e. condi ion (i) holds.
Fo each j∈ {1,...,p−1} A, we ha e j< j+1. Hence, as λjB j, λj+1B j+1 a e
s-hype cyclic, P oposi ion 5.1, (ii) ⇒(iii), and he same app oach as in he p oo
o Co olla y 4.2 in [11] yield |λj|<|λj+1|, i.e. condi ion (ii) holds. Finally, o
each j∈A, we ha e j= j+1. Hence, he s-hype cyclici y o
λjB j, λj+1B j+1 =λj+1
λj
·λjB j
implies |λj+1/λj|= 1 (see P oposi ion 4.1(a)) and hus |λj|=|λj+1|, i.e. condi ion
(iii) holds. 
Fo ins ance, he ope a o s 2B, 3B2,−3B2, being no d-hype cyclic, a e s-
hype cyclic. Fu he s udy o d-hype cyclici y o weigh ed unila e al and bila e al
backwa d shi s can be ound in [9].
6. s-hype cyclici y in spaces o holomo phic unc ions
Le G⊂Cbe a domain, ha is, a nonemp y connec ed open subse o C. We
endow he space H(G) o all holomo phic (o analy ic) unc ions G→Cwi h he
opology o uni o m con e gence on compac a, so ha H(G) becomes a sepa able
F ´eche space. In his sec ion we a e conce ned wi h s-hype cyclici y o ini e se s
o ope a o s on H(G) (o on subspaces o i ) o ce ain domains G.
Recall ha i Xis a opological ec o space and T∈L(X), hen Tis said
o be supe cyclic p o ided ha he e exis s some x0∈Xwhose p ojec i e o bi
{λTnx0:n∈N, λ ∈K}is dense in X. I T1,...,Tp∈L(X), hey a e called d-
supe cyclic (see [7]) i he e is x0∈Xsuch ha {λ[Tn
1,...,Tn
p]x0:n∈N, λ ∈K}
is dense in Xp. Consis en ly, we say ha T1,...,Tpa e s-supe cyclic whene e
{λ[Tn
1,...,Tn
p]x0:n∈N, λ ∈K} ⊃ ∆(Xp).
Le LFT(D) deno e he amily o all linea ac ional ans o ma ions ϕ(z) =
az+b
cz+do he complex plane such ha ϕ(D)⊂D. The sub amily Au (D) o au omo -
phisms o Dconsis s o all on o membe s o LFT(D). See e.g. [27, Chap e 1] o
e minology ela ed o hese amilies. I ν∈R, hen Sνdeno es he weigh ed Ha dy
space Sν={ (z) = Pn≥0anzn∈H(D) : k k:= (Pn≥0|an|2(n+ 1)2ν)1/2<∞}.
Each Sνis a Hilbe space, and he choices ν=−1/2,0,1/2 co espond, espec i-
ely, o he classical Be gman, Ha dy and Di ichle spaces. Thanks o he esul s
in [7], we ob ain wi hou e o he nex wo asse ions.
P oposi ion 6.1. Le ϕ1, . . . , ϕp∈LFT(D)pai wise dis inc . Then he ollowing
a e equi alen :
(a) Cϕ1,...,Cϕpa e s-supe cyclic on H(D).
(b) µ1Cϕ1,...,µpCϕpa e s-mixing on H(D) o all nonze o scala s µ1,...,µp.
(c) Cϕ1,...,Cϕpa e d-supe cyclic on H(D).
(d) µ1Cϕ1,...,µpCϕpa e d-mixing on H(D) o all nonze o scala s µ1,...,µp.
20 BERNAL AND JUNG
(e) ϕ1...,ϕpha e no ixed poin in D, and sa is y ha i any wo ϕl, ϕjha e
he same a ac i e ixed poin α, hen ϕ′
l(α) = ϕ′
j(α)<1is no possible.
P oo . The equi alence o (c), (d) and (e) is p o ed in [7, Theo em 4]. The im-
plica ions (d) ⇒(b) ⇒(a) a e i ial. Finally, (a) ⇒(e) is p o ed in ac in he
p oo o Theo em 4 in [7]. Indeed, i is used he e a esul (Lemma 14 in [7]) as-
se ing ha i ϕ1, ϕ2∈LFT(D) a e hype bolic and sha e an a ac i e ixed poin
αwi h ϕ′
1(α) = ϕ′
2(α), hen Cϕ1, Cϕ2a e no d-supe cyclic on H(D). Bu a close
look a i s p oo shows ha Cϕ1, Cϕ2a e in ac e en no s-supe cyclic; indeed, ia
con adic ion, only one unc ion gis assumed o be simul aneously app oxima ed
by p ojec i e o bi s. 
P oposi ion 6.2. Le ϕ1,...,ϕp∈LFT (D)pai wise dis inc and le ν < 1/2.
Then he ollowing a e equi alen :
(a) Cϕ1,...,Cϕpa e s-supe cyclic on Sν.
(b) Cϕ1,...,Cϕpa e s-mixing on Sν.
(c) Cϕ1,...,Cϕpa e d-supe cyclic on Sν.
(d) Cϕ1,...,Cϕpa e d-mixing on Sν.
(e) Each ϕlis a pa abolic au omo phism o a hype bolic map wi hou ixed
poin s in D, and he e a e no wo ϕl, ϕjha ing a common ixed poin α
such ha ϕ′
l(α) = ϕ′
j(α)<1.
P oo . The equi alence o (c), (d) and (e) is p o ed in [7, Theo em 3]. The impli-
ca ions (d) ⇒(b) ⇒(a) a e i ial. As o (a) ⇒(e), obse e ha in he p oo
o Theo em 3 in [7], only he supe cyclici y o each Cϕlis necessa y o he i s
asse ion in (e) and ha he Compa ison P inciple [7, P oposi ion 8] – ha also
wo ks o s-supe cyclici y– implies ha Cϕ1,...,Cϕpa e s-supe cyclic on H(D).
Now, he second asse ion o (e) ollows om P oposi ion 6.1. 
Rema ks 6.3. 1. Recall ha i Xis an F-space and T∈L(X) is in e ible
and hype cyclic, hen T−1is also hype cyclic. Analogously as in Example 22
in [7], by combining he p eceding wo p oposi ions, we ob ain ha he e a e
hype bolic ϕ1, ϕ2∈Au (D) such ha Cϕ1, Cϕ2a e d-hype cyclic (so s-hype cyclic)
on H2(D) ( he Ha dy space) and on H(D), and Cϕ−1
1= (Cϕ1)−1, Cϕ−1
2= (Cϕ2)−1
a e e en no s-supe cyclic on H2(D) o H(D) (no e ha ϕ−1
1and ϕ−1
2a e also
hype bolic). Hence, in gene al, he d-hype cyclici y o T1,...,Tpdoes no imply
he s-hype cyclici y o T−1
1,...,T−1
pi T1,...,Tpa e in e ible. Mo eo e , ini ely
many composi ion ope a o s gene a ed by non-ellip ic au omo phisms o Dmay
be no s-hype cyclic on H(D) o on H2(D).
2. Fu he s udy o d-hype cyclici y o composi ion ope a o s, his ime on weigh ed
Be gman spaces on D, is pe o med in [30].
In 1929 Bi kho [12] p o ed ha he ansla ion ope a o τa(a∈C {0}) gi en
by (τa )(z) = (z+a) is hype cyclic on he space H(C) o en i e unc ions. I is
SIMULTANEOUS UNIVERSALITY 21
p o ed in [4, P op. 5.5] and [11, Theo em 3.1] ha i a1,...,apa e pai wise dis inc
nonze o complex numbe s, hen τa1,...,τapa e d-hype cyclic. T i ially, we ob ain:
i a1,...,ap∈C {0}, hen τa1,...,τapa e s-hype cyclic. As he nex p oposi ion
shows, we may ob ain a sligh ex ension o weigh ed ansla ion ope a o s.
P oposi ion 6.4. Le p≥2, and le a1,...,ap, λ1,...,λp∈C {0}such ha
|λj|=|λl| o all j, l ∈ {1,...,p}wi h aj=al. Then he e is a sequence (nk)⊂N
such ha he sequences (λ1τa1)nk,...,(λpτap)nka e s-mixing. In pa icula , he
ope a o s λ1τa1,...,λpτapa e densely s-hype cyclic on H(C).
P oo . Selec a ini e sequence {j(1) < j(2) ··· < j(q)} ⊂ {1,...,p}sa is y-
ing ha , i bl:= aj(l)(l= 1,...,q), hen he bl’s a e pai wise dis inc and
{a1,...,ap}={b1,...,bq}. Le µl:= λj(l). Conside he ope a o s Tj:= λjτaj
(j= 1,...,p) and Sl:= Tj(l)=µlτbl(l= 1,...,q).
Le us p o e ha S1, . . . , Sqa e s-mixing. In ac , by ollowing he app oach
o he p oo o [11, Theo em 3.1], we can p o e ha hey a e e en d-mixing.
To his end, and aking in o accoun ha he se s V(h, , ε) := { ∈H(C) :
| (z)−h(z)|< ε o all z∈B(0, )}(h∈H(C), ε > 0, > 0), o m a basis o
he opology o H(C), i is enough o p o e ha , o gi en h, g1,...,gq∈H(C)
and ε, > 0, he e is n0∈Nsuch ha , o e e y n≥n0, he e exis s an en i e
unc ion wi h
| (z)−h(z)|< ε and |(Sn
l )(z)−gl(z)|< ε (z∈B(0, ), l = 1,...,q).(1)
Selec n0∈Nwi h n0>maxi6=l2
|bi−bl|+ max1≤l≤q2
|bl|. Then, o each n≥n0, he
disks B(0, ), B(nb1, ), . . . , B(nbq, ) a e pai wise disjoin . Pick s > such ha
he disks B(0, s), B(nb1, s), . . . , B(nbq, s) a e s ill pai wise disjoin . Le K:=
B(0, )∪B(nb1, )∪ · · · ∪ B(nbq, ) and Ω := B(0, s)∪B(nb1, s)∪ · · · ∪ B(nbq, s).
No e ha Ω is an open se , Ω ⊃Kand Kis a compac subse ha ing connec ed
complemen . Conside he unc ion F: Ω →Cde ined by
F(z) = h(z) i z∈B(0, s) and F(z) := µ−n
lgl(z−nbl) i z∈B(nbl, s) (1 ≤l≤q).
Then F∈H(Ω). F om Runge’s app oxima ion heo em (see e.g. [16]), i ollows
ha he e exis s a polynomial (so ∈H(C)) such ha | (z)−F(z)|< ε/(1 +
|µn
l|) o all z∈K. Bu his implies ha | (z)−h(z)|< ε on B(0, ) and
|µn
l (z)−gl(z−nbl)|< ε on B(nbl, ). Since he las inequali y is equi alen o
|µn
l (z+nbl)−gl(z)|< ε on B(0, ), (1) is ob ained.
As he se D:= {λj/λl:j, l ∈ {1,...,p}wi h aj=al} ⊂ Tis ini e, i is
a Di ichle se . Then he e is a s ic ly inc easing sequence (nk)⊂Nsuch ha
ξnk→1 as k→ ∞, o all ξ∈D.
Fix a subsequence (mk) o (nk). Since S1,...,Sqa e s-mixing, he se
s-HC((Smk
1),...,(Smk
q)) is dense (see P oposi ion 3.3). Fix in
s-HC((Smk
1),...,(Smk
q)). Fo each ν∈ {1,...,p} he e is a unique l=l(ν)∈

22 BERNAL AND JUNG
{1,...,q}such ha aν=bl, so ha |λν|=|µl|. Obse e ha ξν:= λν/µl∈D.
Then ξnk
ν→1, hence ξmk
ν→1 (k→ ∞) o all ν∈ {1,...,p}. Gi en g∈H(C),
we can ind a subsequence (pk) o (mk) wi h Spk
l(ν) →g(k→ ∞) uni o mly
on compac a o e e y ν∈ {1,...,p}. Since ξpk
ν→1 o all ν, we ob ain ha
Tpk
ν =ξpk
νSpk
l(ν) −→ 1·g=g(k→ ∞) uni o mly on compac a o e e y
ν= 1,...,p. The e o e ∈s-HC((Tmk
1),...,(Tmk
p)), which shows ha his se
is dense. By P oposi ion 3.3, he sequences (Tnk
1),...,(Tnk
p) a e s-mixing, as e-
qui ed. 
Ano he impo an collec ion o ope a o s on H(C) is ha o di e en ia ion
ope a o s. Conside he de i a i e ope a o D: ∈H(C)7→ ′∈H(C). I s
hype cyclici y on H(C) was p o ed by MacLane in 1952 [25]. I is shown in
[11, P op. 3.3] ha i p≥2, 1,..., p∈Nwi h 1<··· < pand λ1,...,λp∈
C {0}, hen λ1D 1,...,λpD pa e d-mixing, so densely d-hype cyclic. Conce ning
s-hype cyclici y, he ollowing p oposi ion shows ha somewha so e assump ions
a e allowed, al hough, simila ly o he las p oposi ion, we ha e no been able o
ob ain he s-mixing p ope y o he whole sequences.
P oposi ion 6.5. Le 1≤ · · · ≤ pbe posi i e in ege s and λ1,...,λp∈C {0},
whe e p≥2. Suppose ha |λj|=|λl| o all j, l ∈ {1,...,p}wi h j= l.
Then he e is a sequence (nk)⊂Nsuch ha he sequences (λ1D 1)nk,...,(λpD p)nk
a e s-mixing. In pa icula , he ope a o s λ1D 1,...,λpD pa e densely s-hype cyclic
on H(C).
P oo . As he se {λj/λl:j, l ∈ {1,...,p}wi h j= l} ⊂ Tis ini e, i is a
Di ichle se . Then he e is a s ic ly inc easing sequence (nk)⊂Nsuch ha
(λj/λl)nk→1 as k→ ∞, o all j, l ∈ {1,...,p}wi h j= l. Pu X0:=
{polynomials}= span{zm:m∈N0}and W0:= ∆(Xp
0). Then X0is dense in
X:= H(C) and W0= ∆(Xp
0)⊃∆(Xp). Le Tj:= λjD j(1 ≤j≤p). Fo each
k∈N, de ine he map Rk:W0→X ia
Rk(zm, . . . , zm) :=
p
X
l=1
1
τ(l)·1
λnk
l
·zm+ lnk
(m+ 1)(m+ 2) ···(m+ lnk),
whe e τ(l) := ca d {i∈ {1,...,p}: i= l}(1 ≤l≤p). Then Rkis ex ended o
he whole W0by linea i y. We ha e:
(i) Tnk
jzm= 0 as soon as nk j> m, so Tnk
jzm→0 as k→ ∞ o all j∈
{1,...,p}and all m≥0. The e o e, by linea i y, Tnk
j→0 (k→ ∞) on X0
o all j∈ {1,...,p}.
SIMULTANEOUS UNIVERSALITY 23
(ii) Fix m∈N0and a compac se K⊂C. The e is M∈(0,+∞) wi h
K⊂B(0, M). Gi en k∈N, we ob ain
sup
z∈K
|Rk(zm,...,zm)| ≤
p
X
l=1
1
τ(l)·1
λnk
l
·Mm+ lnk
(m+ 1)(m+ 2) ···(m+ lnk)
≤
p
X
l=1
1
τ(l)
Mm+ lnk/λnk
l
(m+ 1)(m+ 2) ···(m+nk)
≤
p
X
l=1
Mm
τ(l)
(M l/λl)nk
nk!→0 (k→ ∞)
Hence, by linea i y, Rk→0 (k→ ∞) poin wise on W0.
(iii) Fix m∈N0,j∈ {1,...,p}and k∈Nwi h nk> m. Le us compu e
he ac ion o Tnk
jRkon each (zm,...,zm). This yields h ee sums, he i s
o hem co esponding o hose l∈ {1,...,p}wi h l< j, ha equals 0.
The e o e
Tnk
jRk(zm,...,zm) = 0 +
p
X
l=1
l= j
1
τ(l)·λj
λlnk·zm
+
p
X
l=1
l> j
1
τ(l)·λj
λlnk·zm+( l− j)nk
(m+ 1)(m+ 2) ···(m+ ( l− j)nk)
−→ 1
τ(j)·zm·
p
X
l=1
l= j
1 + 0 = zm(k→ ∞)
uni o mly on compac a in C, because τ(j) = τ(l) and (λj
λl)nk→1 o all
(j, l) wi h j= l. By linea i y again, we ge Tnk
jRk(w,...,w)→w o all
j= 1,...,p and all (w,...,w)∈W0.
The conclusion now ollows om Theo em 3.7 (o om Rema k 3.8.1). 
Fo ins ance, he ope a o s 5D, D2,−D2, eiD2,1
10D3,−3D4a e s-hype cyclic,
bu clea ly no d-hype cyclic.
An ex ension uni ying bo h Bi kho ’s and MacLane’s heo ems akes place by
conside ing con olu ion ope a o s on H(C), ha is, ope a o s commu ing wi h
all ansla ions τa. Le Φ(z) = P∞
n=0 anzn∈H(C). Then Φ is said o be
o exponen ial ype p o ided ha he e a e posi i e cons an s A, B such ha
|Φ(z)| ≤ Aexp(B|z|) o all z∈C. Then i s associa ed di e en ial ope a o
Φ(D) = P∞
n=0 anDngi en by Φ(D) =P∞
n=0 an (n)( ∈H(C)) de ines an ope-
a o on H(C). Mo eo e , an ope a o T∈L(H(C)) is o con olu ion i and only
i T= Φ(D) o some en i e unc ion Φ o exponen ial ype. No e ha Dand τa
24 BERNAL AND JUNG
a e special cases ( ake Φ(z)≡zand Φ(z)≡eaz, esp.). Gode oy and Shapi o
[17] p o ed in 1991 ha any nonscala con olu ion ope a o is hype cyclic. I G
is any domain in C, hen Φ(D) is also an ope a o on H(G) whene e Φ is o
subexponen ial ype, ha is, o gi en ε > 0 he e is a cons an A > 0 such ha
|Φ(z)| ≤ Aexp(ε|z|) o all z∈C. We ha e ha also Φ(D) is hype cyclic on
H(G) p o ided ha Gis simply connec ed (i.e. i s complemen wi h espec o
he one-poin compac i ica ion C∞o Cis connec ed) and Φ is no cons an . Fo
s-hype cyclici y, we p esen he ollowing asse ion, wi h which we pu an end o
his in oduc o y pape on s-uni e sali y.
P oposi ion 6.6. Assume ha G⊂Cis a simply connec ed domain and ha
Φ1,...,Φpa e en i e unc ions o subexponen ial ype (o jus o exponen ial ype
i G=C). Assume also ha he se
U0:= λ∈C: max
1≤j≤p|Φj(λ)|<1
is nonemp y and ha each se
Ui:= λ∈C:|Φi(λ)|>1 and max
1≤j≤p|Φj(λ)| ≤ |Φi(λ)|(1 ≤i≤p)
has nonemp y in e io U0
i. Suppose, in addi ion, ha whene e i, j ∈ {1,...,p}
sa is y |Φi(λ)|=|Φj(λ)| o some λ∈U0
i, he e exis s ζ∈Twi h Φj=ζ·Φi.
Then he e is a sequence (nk)⊂Nsuch ha he sequences (Φ1(D))nk,...
. . . , (Φp(D))nka e s-mixing. In pa icula , he ope a o s Φ1(D),...,Φp(D)a e
densely s-hype cyclic on H(C).
P oo . We w i e eλ:= exp(·λ)|G o λ∈C. I is easy o see ha he unc ions eλ
a e linea ly independen . Deno e Vi:= U0
i(1 ≤i≤p). As U0, V1,...,Vpa e open
and nonemp y, we ob ain ha X0:= span{eλ:λ∈U0}is dense in X:= H(G)
(because Gis simply connec ed: use Runge’s app oxima ion heo em oge he
wi h he ac ha span{exp(·λ) : λ∈U0}is dense in H(C); see e.g. [17, Sec . 5]).
Hence W0:= Qp
i=1 span{eλ:λ∈Vi}is dense in Xp.
As A:= {ζ∈T: exis l, j ∈ {1,...,p}wi h Φj=ζΦl} ⊂ Tis ini e, i is
a Di ichle se ; hence he e is a s ic ly inc easing sequence (nk)⊂Nsuch ha
ζnk→1 o all ζ∈A.
Fo each i∈ {1, . . . , p}, we pu Ti:= Φi(D)|H(G),Ei:= {j∈ {1,...,p}: exis s
ζ∈Twi h Φj=ζΦi}and τ(i) := ca d(Ei). No ice ha i i∈Ej, hen Ei=Ej
(jus use ha Tis a mul iplica i e g oup), hence τ(i) = τ(j). Gi en i∈ {1,...,p}
and i∈span{eλ:λ∈Vi}, he e a e uniquely de e mined scala s ci,1,...,ci,J(i)∈
Cand pai wise dis inc λi,1,...,λi,J(i)∈Visuch ha i=PJ(i)
l=1 ci,leλi,l . Fo k∈N
we de ine Rk:W0→Xas
Rkw:=
p
X
i=1
1
τ(i)·
J(i)
X
l=1
ci,l
Φi(λi,l)nk·eλi,l ,(1)
SIMULTANEOUS UNIVERSALITY 25
whe e w= ( 1,..., p)∈W0and he i’s a e as abo e. We ha e:
(i) I λ∈U0and j∈ {1,...,p}, hen Tnk
jeλ= Φj(λ)nkeλ→0 as k→ ∞,
because |Φj(λ)|<1. By linea i y, we ge Tnk
j→0 on X0.
(ii) Le w= ( 1,..., p)∈W0, so ha i=PJ(i)
l=1 ci,leλi,l , as abo e. Since
|Φi(λi,l)|>1, we ge |Φi(λi,l)nk| → +∞as k→ ∞, o each i∈ {1,...,p}
and each l= 1, . . . , J(i). F om (1) one de i es ha Rkw→0.
(iii) Again, le w= ( 1,..., p)∈W0, wi h i=PJ(i)
l=1 ci,leλi,l . Fix j∈
{1,...,p}and k∈N. We compu e
Tnk
jRkw=
p
X
i=1
1
τ(i)·
J(i)
X
l=1
ci,l
Φi(λi,l)nk·Tnk
jeλi,l
=
p
X
i=1
1
τ(i)·
J(i)
X
l=1
ci,l ·Φj(λi,l)
Φi(λi,l)nk
eλi,l =Ak+Bk,
whe e Ak(Bk, esp.) deno es he pa o he p eceding sum co esponding
o hose i∈Ej(i6∈ Ej, esp.). I i∈Ej, he e is ζ=ζi,j ∈Asuch ha
Φj=ζ·Φi, so ha Φj(λi,l)
Φi(λi,l)nk=ζnk→1 as k→ ∞. No e ha τ(i) = τ(j)
i i∈Ej. The e o e, on he one hand,
Ak→
p
X
i=1
i∈Ej
1
τ(i)·
J(i)
X
l=1
ci,l ·eλi,l =1
τ(j)·
p
X
i=1
i∈Ej
J(i)
X
l=1
ci,l ·eλi,l =1
τ(j)·
p
X
i=1
i∈Ej
i.
On he o he hand, i i6∈ Ej, we ha e ha |Φj(λi,l)/Φi(λi,l)|<1 o all
l= 1,...,J(i) (indeed, as λi,l ∈Vi, we ha e |Φj(λi,l)| ≤ |Φi(λi,l)|; i we
assume |Φj(λi,l)|=|Φi(λi,l)|, hen he e would exis ζ∈Twi h Φj=ζ·Φi,
which would yield i∈Ej, a con adic ion). Hence Φj(λi,l)
Φi(λi,l)nk→0, so
Bk→0. This en ails
Tnk
jRkw=Ak+Bk→1
τ(j)·
p
X
i=1
i∈Ej
i(k→ ∞),
and he las ec o belongs o con ({ 1,..., p}) since in he las sum he e
a e exac ly τ(j) summands.
The conclusion ollows, once again, om he s-hype cyclici y c i e ion (Theo em
3.7). 
Rema k 6.7. P oposi ion 3.4 in [11] (see also [4, Theo em 5.3]) asse s ha i U0
and Wi:= {λ∈C:|Φi(λ)|>1 and maxj6=i|Φj(λ)|<|Φi(λ)|} (1 ≤i≤p) a e
nonemp y, hen Φ1(D),...,Φp(D) a e d-mixing. I hese assump ions a e sa is ied,
hen he assump ions o P oposi ion 6.6 a e also sa is ied. No e ha P oposi ion
6.6 includes he case Φ1= Φ, Φj=cjΦ wi h |cj|= 1 (j= 2,...,p).