Sequences of differential operators: exponentials, hypercyclicity and equicontinuity

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. Re e ences [Abe] Y. Abe, Uni e sal holomo phic unc ions in se e al a iables, Analysis 17 (1997), 71–77. [AbZ] Y. Abe and P. Zappa, Uni e sal unc ions in complex gene al g oups, J. Ap- p ox Theo y 100 (1999), 221–232. [A G] D.H. A mi age and P.M. 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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