Sequences of differential operators: exponentials, hypercyclicity and equicontinuity

Sequences of differential operators: exponentials, hypercyclicity and equicontinuity by L. BERNAL–GONZ´ ALEZ and J.A. PRADO–TENDERO Abstract In this paper, an eigenvalue criterion for hypercyclicity due to the first author is improved. As a consequence, some new sufficient conditions for a sequence of infinite order linear differential operators to be hypercyclic on the space of holomorphic functions on certain domains of C Nare shown. Moreover, several necessary conditions are furnished. The equicontinuity of a family of operators as before is also studied, and it is even characterized if the domain is C N. The results obtained extend or improve earlier work of several authors. Key words and phrases: hypercyclic operators and sequences, equicontinuous family, infinite order linear differential operator, subexponential and exponential type, eigenvalue criterion, total subset, exponential functions, Runge domain, polydomain. 2000 Mathematics Subject Classification: Primary 47B38. Secondary 30E10, 47A16, 47E05, 47F05. This work has been supported in part by D.G.E.S. PB96–1348 and the Junta de Andaluc´ıa. 1 Introduction, notation and preliminary results. Throughout this paper we denote by N the set of positive integers, by R the real line, by C the field of complex numbers, and by N 0the set N 0= N ∪{0}. Let X, Y be two linear topological spaces, Ti:X→Y(i∈I:= an arbitrary index set) a family of continuous linear mappings, and x∈X. Then xis said to be hypercyclic or universal for (Ti) whenever its orbit {Tix:i∈I}under (Ti) is dense in Y. The family (Ti) is called hypercyclic whenever it has a hypercyclic vector. Note that if (Ti) is hypercyclic then it is not equicontinuous, but the converse is false in general. In the case I= N , it is clear that, in order that a sequence (Tn) can be hypercyclic, Ymust be separable. If T:X→Xis an operator (= continuous linear selfmapping) on X, then a vector x∈Xis said to be hypercyclic for Tif and only if it is hypercyclic for the sequence (Tn) of iterates of T, i.e., Tn=T◦T◦· · ·◦T (n–fold). The operator Tis hypercyclic when there is a hypercyclic vector for T. The symbols HC(T) and HC((Ti)) will denote, respectively, the set of hypercyclic vectors of an operator Tand of a family Ti:X→Y(i∈I) of continuous 1 linear mappings. In the last two decades an extensive literature about the topic of hypercyclicity has been developed; a good survey for the whole history is [Gr1]. Let Gbe a nonempty open subset of C N(N∈ N ). We say that Gis a domain when, in addition, it is connected. A domain G⊂ C Nis said to be a Runge domain (see [Hor] or [Kra]) if and only if each holomorphic function on Gcan be uniformly approximated by polynomials on compact subsets of G. Note that, if N= 1, then Gis a Runge domain if and only if it is simply connected. By H(G) we denote, as usual, the Fr´echet space of holomorphic functions on G, endowed with the compact-open topology. Recall that the family {V(K, ε) : ε > 0, K is a compact subset of G}is a neighbourhood basis for the origin in H(G). Here V(K, ε) := {f∈H(G) : ||f||K< ε}. For A⊂ C Nwe have denoted ||g||A:= sup{|g(z)|:z∈A}whenever gis a complex function defined on the set A. G. Godefroy and J.H. Shapiro [GoS, Section 5] proved in 1991 the following generalization of the classical approximation theorems by translates and derivatives of a single entire function due respectively to Birkhoff [Bir] and MacLane [Mac]: If Tis an operator on the space H( C N) of entire functions on C Nthat commutes with each of the translation operators τa(a∈ C N) given by τaf(z) = f(z+a) (f∈H( C N), z ∈ C N), and is not a scalar multiple of the identity, then HC(T) is a dense Gδ-subset of H( C N); in addition, HC(T) contains all nonzero functions of a dense, T–invariant, linear submanifold of X:= H( C N). P. Bourdon [Bou] and D. Herrero [Her] proved independently that every hypercyclic operator Ton any Banach space X(in fact, on any real or complex locally convex space X; see [Ans] and [Bes]) has the same property. The first author of the present paper [Be4] has recently shown that if Xand Yare two separable metrizable linear topological spaces and if Tn:X→Y(n∈ N ) is a sequence of continuous linear mappings for which there is an increasing sequence (nj) of positive integers with the property that HC((Tmj)) is dense for every subsequence (mj) of (nj), then HC((Tn)) ∪ {0} contains a dense linear submanifold of X. Given N∈ N , denote by Dj(1 ≤j≤N) complex partial differentiation with respect to the j-th coordinate. A multi–index is an N–tuple p= (p1, ..., pN) of nonnegative integers. Denote |p|=p1+· · · +pN,p! = p1!· · · pN!, Dp=Dp1 1◦ · · · ◦ DpN N(with D0 j=I= the identity operator for every j∈ {1, ..., N}), and |z|= (|z1|2+· · · +|zN|2)1/2,zp=zp1 1· · · zpN N,zw =z1w1+· · · +zNwNif z= (z1, ..., zN), w= (w1, ..., wN). An entire function Φ(z) = P|p|≥0apzpis said to be of exponential type whenever there exist positive constants Aand Bsuch that |Φ(z)| ≤ AeB|z| 2 (z∈ C N). For later references, we denote by Ethe class of all entire functions of exponential type. An entire function Φ is said to be of subexponential type if and only if, given ε > 0, there is a positive constant A=A(ε) such that |Φ(z)| ≤ Aeε|z| (z∈ C N). Every entire function of subexponential type is obviously in E. It is easy to realize (see, for instance, [Val], [Dic] or [Be3]) that if G⊂ C Nis a nonempty open subset and Φ is an entire function as above with subexponential type, then the series Φ(D) = P|p|≥0apDpdefines an operator on H(G). If G= C N, the same result holds just by assuming that Φ is of exponential type. So Φ(D) defines, under the latter conditions, an infinite order linear differential operator with constant coefficients. It is shown in [GoS] that, given an operator Lon H( C N), then L commutes with every translation operator τa(a∈ C N) if and only if Lcommutes with each Dk(1 ≤k≤N) if and only if L= Φ(D) for some entire function Φ in E. As a consequence of an eigenvalue criterion for hypercyclicity [Be3, Theorem 7], the first author obtained some extensions of Godefroy–Shapiro’s result [Be3, Theorems 8–9], this time about the hypercyclicity of a sequence of operators (Φn(D)) defined on the space of holomorphic functions on a Runge domain Gof C N. Furthermore, conditions about the equicontinuity of a sequence (cnDn), where (cn)⊂ C (note that this is the special case Φn(z) = cnzn), are shown in [Be1] and [Be2] (see also [Cal], when each cnis replaced to a holomorphic fuction cn(z)). Our aim in this paper is to provide with a more general eigenvalue criterion and, as a consequence, new sufficient conditions for the hypercyclicity of a sequence of infinite order linear differential operators. In addition, necessary conditions are established, and some special cases are analyzed. Necessary conditions and sufficient conditions for its equicontinuity are also furnished, and in particular we characterize completely the equicontinuity in H( C N). 2 Eigenvalues, exponentials, hypercyclicity and equicontinuity. Likewise in [GoS, Section 5] and [Be3, Theorems 8–9], the key of the proof of hypercyclicity is to provide a good supply of eigenvectors of the corresponding operators. Recall that, in a linear topological space, a subset is said to be total whenever its linear span is dense. If Tis an operator and eis an eigenvector, then 3 we denote by λ(T, e) its corresponding eigenvalue. Next, we state as a lemma the following rather general hypercyclicity criterion, which can be found in [Gr1]. Lemma 2.1 Assume that Xis a Baire topological vector space, Yis a separable metrizable topological vector space and Tn:X→Y(n∈ N )are continuous linear mappings. Suppose that there are dense subsets X0of Xand Y0of Yand mappings Sn:Y0→Xsuch that (a) for every x∈X0, there exists an increasing sequence (nk) = {n1< n2< ...} of positive integers with Tnkx→0 (k→ ∞), (b) for every y∈Y0,(Sny)converges, and (c) for every y∈Y0,Tn(Sny)→y(n→ ∞). Then HC((Tn)) is residual. As noted in [Gr1, Remark 2], if all the limits in (b) are zero then we may weaken (a) to be for every x∈X0, there exists an increasing sequence (nk) = {n1< n2< ...}of positive integers such that (Tnkx)converges. Furthermore, the quantifier “∃(nk)” can be shifted from (a) to (b) or (c). Under the same hypothesis for Xand Y, it can be proved (see, for instance, [Be2]) that the following condition is also sufficient in order that HC((Tn)) be residual: there exist dense subsets X0of Xand Y0of Ysatisfying that for every x∈X0and every y∈Y0there exists an increasing sequence (nk) of positive integers and a sequence (xk)⊂Xsuch that xk→0, Tnkx→0 and Tnkxnk→yas k→ ∞. By using the latter result, the next eigenvalue criterion can be proved (see [Be3, Theorem 7]): Let Xbe a separable F–space and (Tn) a sequence of operators on X. Assume that there are two total subsets A,Bof Xsatisfying that for every pair of finite subsets F1⊂ A and F2⊂ B there is an increasing sequence (nk) in N such that every element in F1∪ F2is an eigenvector for each Tnkin such a way that λ(Tnk, a)→0 (k→ ∞) for all a∈ F1and λ(Tnk, b)→ ∞ (k→ ∞) for all b∈ F2. Then HC((Tn)) is residual. If we employ Lemma 2.1 (and the note after it) instead of the just mentioned result then the following eigenvalue criterion can be obtained. The proof is left to the interested reader. 4 Theorem 2.2 Let Xbe a separable F–space and (Tn)be a sequence of operators on X. Assume that there are two total subsets A,Bof Xsatisfying at least one of the following conditions: (A) For every finite subset F ⊂ A there is an increasing sequence (nk)in N such that every element in Fis an eigenvector for each Tnkin such a way that λ(Tnk, a)→0 (k→ ∞)for all a∈ F. In addition, every element in Bis an eigenvector for each Tnin such a way that for every b∈ B the sequence (λ(Tn, b)) converges to a nonzero scalar. (B) For every finite subset F ⊂ A there is an increasing sequence (nk)in N in such a way that for every a∈ F the sequence (λ(Tnk, a)) converges. In addition, every element in Bis an eigenvector for each Tnin such a way that, for every b∈ B,(λ(Tn, b)) → ∞ (n→ ∞). (C) Every element in Ais an eigenvector for each Tnin such a way that λ(Tn, a)→ 0 (n→ ∞)for every a∈ A. In addition, for every finite subset F ⊂ B there is an increasing sequence (nk)in N such that every element in Fis an eigenvector for each Tnkin such a way that for every b∈ F the sequence (λ(Tnk, b)) converges to a nonzero scalar. (D) Every element in Ais an eigenvector for each Tnin such a way that for every a∈ A the sequence (λ(Tn, a)) converges. In addition, for every finite subset F ⊂ B there is an increasing sequence (nk)in N such that every element in F is an eigenvector for each Tnkin such a way that (λ(Tnk, b)) → ∞ (k→ ∞) for every b∈ F. Then HC((Tn)) is residual. In other order of ideas, recall that Edenotes the class of entire functions on C N of exponential type. We say that a subset S⊂ C Nis an E–unicity set whenever the following property holds: if f∈ E and f(z) = 0 for all z∈Sthen f≡0. Note that, by the identity principle for holomorphic functions, if fis an arbitrary entire function vanishing at Sand Sis a nonempty open set (or even just a set with at least an accumulation point if N= 1) then f≡0. This is not necessary for the class E; for instance, if N= 1 and χ:= lim supr→∞ log n(r) log r>1, where n(r) is the number of points of S∩ {|z| ≤ r}, then Sis an E–unicity set (e.g., S={n1/2:n∈ N }, which gives χ= 2). Indeed, if f6≡ 0, the latter condition 5 would imply that the convergence exponent of the sequence of zeros of fis strictly greater that the growth order of f, which is clearly impossible. The next lemma will be useful later. Its proof is classical, but we include it for the sake of completeness. If c∈ C Nthen we denote ec(z) = exp(cz). Lemma 2.3 If Sis an E–unicity set then M(S) := {ec:c∈S}is total in H( C N). Proof. Fix a functional L∈H( C N)∗(= the topological dual space of H( C N)) such that L(ec) = 0 for all c∈S. Consider the Laplace transform ˜ Lof L(see [Hor, p. 100]) given by ˜ L(z) = L(ez) (z∈ C N). Then it is easy to show that ˜ L is an entire function on C Nof exponential type which vanishes at S. Since Sis an E–unicity set, we get ˜ L≡0. Then (Dp˜ L)(0) = 0 for all p∈ N N 0. But it is easy to show by induction that (Dp˜ L)(0) = L(αp), where αp(t) = tp(t∈ C N). By linearity, Lvanishes at every polynomial, so L≡0 because the set of polynomials is dense in H( C N). Summarizingly, if L(f) = 0 for all f∈M(S) then L(f) = 0 for all f∈H( C N). By the Hahn–Banach theorem, the linear span of M(S) is dense in H( C N) or, equivalently, M(S) is total. Next, we state here eight conditions that may or may not be satisfied by a sequence (Φn)⊂H( C N). Recall that if Φ(z) = P|p|≥0apzp∈H( C N) and Φ is not identically zero, its multiplicity for the zero at the origin is m(Φ) = min{|p|:ap6= 0}. Note that Φ(D)ec= Φ(c)ecfor all c∈ C N, so ecis an eigenvector of Φ(D) with eigenvalue Φ(c). (P) There are two E–unicity sets A, B in C Nsuch that for every pair of finite subsets F1⊂Aand F2⊂Bthere exists an increasing sequence (nk)⊂ N with Φnk(a)→0 (k→ ∞) for all a∈F1and Φnk(b)→ ∞ (k→ ∞) for all b∈F2. (Q) There is an E–unicity set Bin C Nsuch that for every finite subset F⊂B there exists an increasing sequence (nk)⊂ N with m(Φnk)→ ∞ (k→ ∞) and Φnk(b)→ ∞ (k→ ∞) for all b∈F. (R) There are two E–unicity sets A, B in C Nsuch that for every finite subset F⊂Athere exists an increasing sequence (nk)⊂ N with Φnk(a)→0 (k→ ∞) for all a∈F, and for each b∈Bthe sequence (Φn(b)) converges to a nonzero complex number. 6 (S) There is an E–unicity set Bin C Nsuch that for each b∈Bthe sequence (Φn(b)) converges to a nonzero complex number, and there exists an increasing sequence (nk)⊂ N with m(Φnk)→ ∞ (k→ ∞). (T) There are two E–unicity sets A, B in C Nsuch that for every finite subset F⊂Athere exists an increasing sequence (nk)⊂ N satisfying that for every a∈Fthe sequence (Φnk(a)) converges. In addition, Φn(b)→ ∞ (n→ ∞) for every b∈B. (U) There are two E–unicity sets A, B in C Nsuch that Φn(a)→0 (n→ ∞) for all a∈A, and for each finite subset F⊂Bthere exists an increasing sequence (nk)⊂ N satisfying that for every b∈Fthe sequence (Φnk(b)) converges to a nonzero complex number. (V) There is an E–unicity set Bin C Nsuch that for each finite subset F⊂B there exists an increasing sequence (nk)⊂ N satisfying that for every b∈F the sequence (Φnk(b)) converges to a nonzero complex number. In addition, m(Φn)→ ∞ (n→ ∞). (W) There are two E–unicity sets A, B in C Nsuch that for every a∈Athe sequence (Φn(a)) converges, and for every finite subset F⊂Bthere is an increasing sequence (nk)⊂ N with Φnk(b)→ ∞ (k→ ∞) for all b∈F. We are now ready to state our next result. In the remaining of this paper, Φ and Φi(i∈I:= an arbitrary index set) will denote entire functions of subexponential type if G6=CNand of exponential type if G= C N,Gbeing a given domain in C N. Thus, the operators Φ(D), Φi(D) (i∈I) are well defined on H(G). Theorem 2.4 Suppose that Gis a Runge domain of C Nand that (Φn)satisfies at least one of the conditions (P)–(W). Then HC((Φn(D))) is residual in H(G). Proof. Recall that, by Lemma 2.3, the set M(S) is total in H( C N) (hence in H(G), because Gis Runge) whenever Sis an E–unicity set. Recall also that the set {zp:p∈ N N o}is total in H(G), because that set spans {polynomials}. Take X=H(G) and Tn= Φn(D) (n∈ N ). Then: Apply the result mentioned just before Theorem 2.2 on A=M(A), B=M(B) if (Φn) satisfies (P), and on A={zp:p∈ N N 0},B=M(B) if (Φn) satisfies (Q). Apply condition (A) of Theorem 2.2 on A=M(A), B=M(B) if (Φn) satisfies (R), and on A={zp: p∈ N N 0},B=M(B) if (Φn) satisfies (S). Apply condition (B) of Theorem 2.2 7 on A=M(A), B=M(B) if (Φn) satisfies (T). Apply condition (C) of Theorem 2.2 on A=M(A), B=M(B) if (Φn) satisfies (U), and on A={zp:p∈ N N 0}, B=M(B) if (Φn) satisfies (V). Finally, apply condition (D) of Theorem 2.2 on A=M(A), B=M(B) if (Φn) satisfies (W). Let us furnish several examples that illustrate Theorem 2.4. The reader will realize that none of the examples below can be derived from Theorems 8, 9 of [Be3]. But before this we should fix some subsets. Consider S={n1/2:n∈ N } and let (rj) be any sequence of positive real numbers such that the plane disks {|z−j1/2|< rj}(j∈ N ) be pairwise disjoint, for instance, rj= 1/6j. Define the compacts sets Kn:= (Ln∪S)∩In(n∈ N ), where In:= [−n, n]×[−n, n] and Ln:= C \[((0,+∞)×(−1/n, 0)) ∪ ∞ [ j=1 {|z−j1/2|< rj/n}]. It is easy to see that each Knhas connected complement. Define the functions fn, gn:Kn→ C (n∈ N ) as fn(z) =    1 (z∈Ln∩In) n(z∈S∩In) and gn(z) =    1 (z∈Ln∩In) 0 (z∈S∩In). It is clear that every fnand every gnis holomorphic on some open subset containing Knand depending on n. Then Runge’s theorem guarantees the existence of polynomials Pn, Qnsatisfying ||Pn−fn||Kn<1/n and ||Qn−gn||Kn<1/n (n∈ N ). Since Ln∩In(S∩In) grows up to C \S(up to S, respectively) as ntends to infinity, the latter two inequalities lead us to the following facts of point convergence: Pn→1 on C \S,Pn→ ∞ on S,Qn→1 on C \Sand Qn→0 on Sas n→ ∞. EXAMPLE 1. There is a residual set of entire functions fon C such that each entire function can be locally uniformly approximated by entire functions of the form n X j=0 Ajnf(j)(n∈ N ), 8 where Ann = 1 and Ajn = (−1)n−jX 1≤i1<i2<···<in−j≤n (i1· · · in−j)1/2(0 ≤j≤n−1). Indeed, it suffices to apply the latter theorem with condition (P) or (T) on A=S, B= C \S, Φn(z) = n Y j=1 (z−j1/2) (n∈ N ) (use Cardano–Vieta’s relations). EXAMPLE 2. The set HC((Pn(D))) is residual in H( C ) because Theorem 2.4 can be applied with condition (T) or (W) on A= C \S,B=S. EXAMPLE 3. The set HC((Qn(D))) is residual in H( C ) because Theorem 2.4 can be applied with condition (R) or (U) on A=S,B= C \S. Analogous properties to (P)–(W) regarding the densely hereditary hypercyclicity of (Φn(D)) can be formulated as in [Be4, Section 3]. This would yield sufficient conditions for the existence of dense (Φn(D))–hypercyclic linear submanifolds in H(G). In his paper, Birkhoff [Bir] essentially proved that given an unbounded sequence (an)⊂ C there exists an entire function in C such that the set of translates {f(z+an) : n∈ N }is dense in H( C ), i.e., the sequence (τan) is hypercyclic (as a matter of fact, the sequence (an) depended on the particular entire function to be approximated; in [Luh] this dependence is dropped). His constructive proof can be adapted to C N: see, for instance, [Abe] and [AbZ]; see also [ArG] for corresponding results for harmonic functions on R N. As a quick application of the latter theorem, we will obtain this Birkhoff theorem in several variables. Theorem 2.5 Assume that S⊂ C N. Then the following conditions are equivalent: (a) Sis unbounded. (b) The family of operators (τa)a∈Sis hypercyclic on H( C N). (c) HC((τa)a∈S)is residual in H( C N). (d) (τa)a∈Sis not equicontinuous on H( C N). Proof. The implications (c) ⇒(b) ⇒(d) are trivial. If Sis bounded, take M∈(0,+∞) with |a| ≤ Mfor all a∈S. Given a basic neighbourhood V(K, ε) for the origin in H( C N), it is clear that [ a∈S τa(V(L, δ)) ⊂V(K, ε), 9 entire function with subexponential type, so part (a) of Theorem 2.9 yields the desired result. For G= C Nwe are able to characterize the equicontinuous families of differential operators. Theorem 2.11 The family of operators {Φi(D) : i∈I}is equicontinuous on H( C N)if and only if (Φi)admits a majorant entire function of exponential type. Proof. The part “only if” is due to Theorem 2.9(b). As for the converse, we can follow step by step the proof of part (a) of Theorem 2.9 with the sole exception that we may choose the polycycle γfar enough from the compact set K(so µcan be choosen as large as desired) in such a way that sup |p|>0, i∈I (p!|cpi|)1/|p|≤µ/2. The constant Mmay be choosen as M= max {1,supi∈I|c0i|}. The proof is finished. In [Be2, Theorem 1] it has been established that if G⊂ C is a simply connected domain and (cn) is a complex sequence with R(G)≤lim sup n→∞ (n!|cn|)1/n then HC((cnDn)) is residual in H(G). A slight generalization can be obtained in the N–dimensional case. The proof is very similar to the 1–dimensional one, so we omit it. Theorem 2.12 Assume that G⊂ C Nis a Runge domain and that (p(n)) is a sequence of multi–indexes with |p(n)|→∞(n→ ∞). If (cn)is a complex sequence with R(G)≤lim sup n→∞ (p(n)! |cn|)1/|p(n)| then the set HC((cnDp(n))) is residual in H(G). As a consequence of Theorems 2.11, 2.12 we can get a characterization of equicontinuity and hypercyclicity of the same sequence in H( C N). This is achieved in the next result, which in turn is an N–dimensional extension of [Be2, Theorem 4] (see also [Be1]). Theorem 2.13 Assume that (cn)is a complex sequence and that (p(n)) is a sequence of nonzero multi–indexes such that |p(n)|→∞(n→ ∞). Then the following properties are equivalent: 16 (a) The sequence ((p(n)! |cn|)1/|p(n)|)is bounded. (b) There is no hypercyclic entire function for (cnDp(n)). (c) The set HC((cnDp(n))) is not residual in H( C N). (d) The sequence (cnDp(n))is equicontinuous on H( C N). Proof. It is evident that (b) implies (c) and that (d) implies (b). Since R( C N) = +∞, we obtain from Theorem 2.12 that (c) implies (a). Assume that (a) holds. Then we can apply Theorem 2.11 with I= N and Φn(z) = cnzp(n). Indeed, there is a constant Mwith p(n)!|cn| ≤ M|p(n)|for all n∈ N , hence the function Φ(z) = ∞ X n=1 M|p(n)| p(n)! zp(n)(z∈ C N) is a majorant entire function for (Φn) with exponential type. Then (d) is true and the proof is finished. We point out that in [Gr2, Corollary to Theorem 4] the part about hypercyclicity of [Be2, Theorem 4] is extended for the case N= 1 to sequences of weighted pseudo-shifts in the space H( C ). The part “only if” of [Be2, Theorem 3] is able to be extended in the same way to the N–dimensional case, as the following theorem shows. Theorem 2.14 Let G=G1× · · · × GN⊂ C Nbe a polydomain with Gj6= C (j= 1, ..., N). Assume that (cn)is a complex sequence and that (p(n)) is a sequence of nonzero multi–indexes such that the sequence of operators (cnDp(n))is equicontinuous on H(G). Then lim n→∞ (p(n)! |cn|)1/|p(n)|= 0. Proof. Consider the number α:= lim supn→∞(p(n)!|cn|)1/|p(n)|. By the way of contradiction, assume that α > 0. Fix a point a= (a1, ..., aN)∈G. Then aj∈Gj and there exist points bj∈ C \Gj(j= 1, ..., N) such that |aj−bj|= inf {|aj−t|: t∈ C \Gj}. Denote R= min {|aj−bj|:j= 1, ..., n}>0. Fix r∈(0, R) with R−r < α. Put K=D(a, r). Then Kis a compact subset of G. Let Lbe any compact subset of Gand δa positive number. Let m > 0 be so small that m (inf {|zj−bj|:j∈ {1, ..., N}, z = (z1, ..., zn)∈L∪K})N< δ. 17 Consider the function f(z) = m QN j=1(zj−bj). Then f∈H(G) and, in addition, fbelongs to V(δ, L). Furthermore, for z= (z1, ..., zN)∈G, |(Tnf)(z)|=p(n)! m|cn| QN j=1 |zj−bj|1+pj(n), where p(n)=(p1(n), ..., pN(n)) and Tn=cnDp(n). Since inf {|t−bj|:|t−aj|< r} ≥ R−rfor every j∈ {1, ..., N}, we get sup {|(Tnf)(z)|:z∈K} ≤ p(n)!|cn|m (R−r)N+|p(n)|=m (R−r)N·p(n)!|cn| (R−r)|p(n)|. But p(nk)!|cnk| (R−r)|p(nk)|→ ∞ (k→ ∞) for some increasing sequence (nk)⊂ N , because α > R −r. Hence sup{|Tnf(z)|:z∈K}=∞. Therefore [ n∈ N Tn(V(δ, L)) 6⊂ V(1, K), which implies that (Tn) is not equicontinuous. The proof is finished. Our final result comes back to hypercyclicity and looks slightly different from the others. It puts the emphasis on the first nonzero Taylor coefficient of each Φn. This time the setting is the complex plane C . Observe that MacLane’s theorem is again recovered if we choose Φn(z) = znfor each n∈ N . Theorem 2.15 Assume that (Φn(z) = ∞ X j=0 cjnzj)is a sequence of nonzero entire functions and denote p(n) := m(Φn) (n∈ N ). Assume that the following three conditions are fulfilled: (a) p(n)→ ∞ as n→ ∞. (b) p(n)|cp(n),n|k/p(n)→ ∞ as n→ ∞ for every k∈ N . (c) Each sequence {cj+p(n),n :n∈ N }(j∈ N )is bounded. Then the set HC((Φn(D))) is residual in H(G)for any simply connected domain G⊂ C . Proof. We are trying to apply Lemma 2.1 with X=H(G) = Y,X0= {polynomials}=Y0and Tn= Φn(D) (n∈ N ). If Pis a polynomial then by 18 (a) there exists n0∈ N with p(n)>degree (P) for all n≥n0, hence DjP= 0 for all j≥p(n) (n≥n0). Therefore TnP= 0 eventually and condition (a) of Lemma 2.1 is satisfied. Now fix mand nin N and try to solve the equation Tnf=zm. Observe that Tn= Ψn(D)◦Dp(n), where Ψn(z) = P∞ j=0 ajnzjand ajn =cj+p(n),n, so a0n6= 0 for all n∈ N . Consider the equation Ψn(D)g=zm,(1) where gis a polynomial of degree not greater than m, say, g(z) = Pm k=0 bknzk. It is easy to see that such a polynomial solution exists. Indeed, (1) is equivalent to m X j=0 ajn( m X k=0 bknzk)(j)=zm, which in turn is the same as the system    Pm j=kaj−k,nbjn ·j! k!= 0 (k= 0,1, ..., m −1) a0nbmn = 1. This is a recurrent square system with determinant am+1 0n6= 0, so it has a unique solution (b0n, ..., bmn) and Cramer’s rule yields bkn =1 am+1 0n · m X j=1 Pjkm(a1n, ..., amn)aj 0n(2) for k∈ {0,1, ..., m}, where Pjkm (j= 1, ..., m) are polynomials of mcomplex variables not depending on n. From (c), there is a finite positive constant M, which does not depend on n, such that |Pjkm(a1n, ..., amn)| ≤ M(3) for all k∈ {0,1, ..., m}and all j∈ {1, ..., m}. Hence a solution of Tnf=zmis f(z) = fn(z) = m X k=0 bkn zk+p(n) (k+p(n))! (n∈ N ), where bkn is given by (2). Let us fix R > 1. Then from (3) we obtain for |z| ≤ R that |fn(z)| ≤ (m+ 1) m X j=1 MRm |a0n|m+1−j·Rp(n) p(n)! →0 (n→ ∞) since (b) and Stirling’s formula leads us to (p(n)! |a0n|m+1−j)1/p(n)→ ∞ (n→ ∞), 19 so the terms ot the latter sequence are eventually greater than, for instance, 1/2R. Therefore (fn) tends to zero in H(G). The proof for the case m= 0 is easier and left to the reader. Define Sn(zm) := fn(z) (m∈ N 0;n∈ N ) and extend Snto Y0by linearity. Then it is clear that SnP→0 (n→ ∞) and Tn(SnP) = P→Pas n→ ∞. Consequently, conditions (b) and (c) in Lemma 2.1 are also fulfilled, as required. For instance, there is an entire function fin C with the property that any entire function can be locally uniformly approximated by functions of the form cn(f(n)+f(n+1)) (n∈ N ), where cn=n−n/(log n)1/2. 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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 protected] E–mail: [email protected] 22