Exponential type of hypercyclic entire functions

Exponential type of hypercyclic entire functions L. Bernal–Gonz´alez ∗A. Bonilla † Abstract In this paper the exponential type of hypercyclic entire functions with respect to a sequence (Φn(D)) of differential operators is considered, where every Φnis an entire function of exponential type. We prove that under suitable conditions certain rates of growth are possible for hypercyclicity while others are not. In particular, our statements extend the negative part of a sharp result on growth of D–hypercyclic entire functions due to Grosse-Erdmann, and are related to a result by Chan and Shapiro about the existence of Φ(D)–hypercyclic functions in certain Hilbert spaces of entire functions. 2000 Mathematics Subject Classification: Primary 47A16. Secondary 30E10, 47B38, 47E05. Key words and phrases: entire function, hypercyclic function, infinite order differential operator, growth, exponential type, dense linear manifold. 1 Introduction and notation Throughout this paper Cwill stand for the complex plane. Nis the set of positive integers, N0=N∪ {0}and B(a, r) (B(a, r)) is the euclidean open (closed, respectively) disk with center aand radius r(a∈C, r > 0). Dis the open unit disk. H(C) denotes, as usual, the linear space of holomorphic functions on G, endowed with the compact-open topology. H(C) becomes a Fr´echet space with this topology, so it is a Baire space; it is also separable. An operator on a topological vector space is a continuous linear selfmapping. The differentiation operator Don H(C) is defined as Df =f0. The exponential type of an entire function Φ(z) = P∞ j=0 ajzjon Cis τ(Φ) = inf{µ > 0 : there exists r0=r0(µ)>0 such that M(Φ, r) := max{|Φ(z)|:z∈ B(0, r)}<exp (µr)∀r > r0}. In other words, τ(Φ) = lim sup r→∞ log M(Φ, r) r. To every entire function Φ we can associate a “formal” infinite order differential operator with constant coefficients T= Φ(D), that is, T=P∞ j=0 ajDjwith D0=I= the identity operator. It is easy to prove (see, for instance, [11, Section 5]) that if Φ has finite exponential type then Φ(D) defines an operator ∗Partially supported by DGES Grant PB96–1348 and the Junta de Andaluc´ıa. †Partially supported by DGES Grant PB98-0444. 0 on H(C). In fact, an operator Ton H(C) has the latter form if and only if T commutes with Dif and only if Tcommutes with every translation operator τa, defined as τa(f)(z) := f(z+a) for all f∈H(C) and all z, a ∈C(see [11, Proposition 5.2]). Note that τa= exp (aD) for every a∈C. Let Xand Ybe topological vector spaces and Tn:X→Y(n∈N) continuous linear mappings. Then a vector x∈Xis said to be hypercyclic (or universal) for the sequence (Tn) whenever the orbit {Tnx:n∈N}is dense in Y. (Tn) is hypercyclic whenever there is at least one hypercyclic vector. If X=Yand Tis an operator on X, then Tis said to be hypercyclic whenever the sequence of iterates (Tn) is hypercyclic, and a vector x∈Xis hypercyclic for Twhenever it is hypercyclic for (Tn). We will employ the following version of the so-called Hypercyclicity Criterion, which can be found in Theorem 2 (and Remark 2 after it) of the survey [13]. Theorem 1.1 Let X, Y be topological vector spaces, in such a way that Xis a Baire space and Yis separable metrizable. Assume that Tn:X→Y(n∈N) are continuous linear mappings. Suppose that there are dense subsets X0of X and Y0of Yand (possibly non-linear and discontinuous) mappings Sn:Y0→ Ysuch that (i) for every x∈X0,Tnx→0 (n→ ∞), (ii) for every y∈Y0, there is an increasing sequence (nk)⊂Nsuch that (Snky)converges, (iii) for every y∈Y0,TnSny→y(n→ ∞). Then the set of (Tn)–hypercyclic vectors of Xis residual, that is, its complement is of first category. We will also use the next result which guarantees the existence of dense hypercyclic linear submanifolds for densely hereditarily hypercyclic sequences of linear operators, see [5, Theorem 2]. Theorem 1.2 Let Xand Ybe two separable metrizable topological vector spaces. If Tn:X→Y(n∈N)is a sequence of continuous linear mappings such that for each sequence n1< n2< n3< . . . of positive integers there is a dense subset of hypercyclic vectors for the subsequence (Tnk), then there is a dense linear submanifold M⊂Xsuch that every vector x∈M\ {0}is hypercyclic for (Tn). G. D. Birkhoff showed in 1929 that every τa(a∈C\ {0}) is hypercyclic on H(C) [6], and G. R. MacLane [16] obtained in 1952 the same conclusion 1 for D. As a simultaneous generalization of these statements, G. Godefroy and J. H. Shapiro obtained in 1991 the following result, see [11, Theorem 5.1]: If Φ is a nonconstant entire function with finite exponential type then the subset of hypercyclic entire functions for Φ(D) is residual; the reader is referred to [4] for corresponding statements about sequences of differential operators. On the other hand, K. G. Grosse–Erdmann proved in 1990 the next sharp statement about the growth of hypercyclic functions, see [12]. Theorem 1.3 There is no hypercyclic entire function ffor Dsuch that |f(z)|=O(exp r √r)as |z|=r→ ∞. However, given any function ϕ: (0,∞)→ (0,∞)with ϕ(r)→ ∞ as r→ ∞, the set of D–hypercyclic functions fwith |f(z)|=O(ϕ(r)exp r √r)as |z|=r→ ∞ is dense in H(C). The same result was independently obtained by Shkarin [17]. We point out here that MacLane [16] had already shown that there are D–hypercyclic functions of exponential type 1, while S. M. Duyos-Ruiz [10] noted that no D–hypercyclic function can be of exponential type less than 1. G. Herzog [15] proved the existence of a D–hypercyclic function growing no faster than rexp r as r→ ∞. The reader is referred to [1–3] and [12, Section 6] for corresponding results about harmonic functions on RNand about shift operators on H(C), respectively. (In [3] and [14], even dense linear manifolds consisting, except for zero, of hypercyclic functions, are obtained). Grosse–Erdmann’s theorem cannot be extended in the same way to different operators Φ(D), that is, there are operators Φ(D) with a trivial least-possible rate of growth of hypercyclic functions. In fact, Duyos Ruiz [9] proved in 1983 that there are entire functions with arbitrarily slow non-polynomial growth which are hypercyclic for every fixed translation operator τa(recall that τa= exp (aD)). Nevertheless, if a∈C\ {0}, it is easy to see –by following step by step the proof of [12]– that the same conclusion of Theorem 1.3 holds for aD–hypercyclic functions just by changing exp rto exp (cr), where c= 1/|a|, that is, cis the common modulus of all solutions of the equation |Φ(z)|= 1, Φ being the function az. Note also that Φ(0) = 0 this time. In 1991, Chan and Shapiro [8, Theorem 2.1] strengthened Duyos-Ruiz’s theorem in the following way: τais hypercyclic on E2(γ) for every a∈C\{0} and every entire function γ(z) = P∞ n=0 γnznsatisfying that γn>0 for each n, the sequence of ratios γn+1/γndecreases to zero as nincreases to ∞and the sequence nγn/γn−1is monotically decreasing. Here E2(γ) is the Hilbert space of entire functions f(z) = P∞ n=0 anznfor which ||f|| ≡ (P∞ n=0 γ−2 n|an|2)1/2<∞. In [8, p. 1447] they point out that the proof of their Theorem 2.1 can be modified to give the following result: If Φ(z) is holomorphic in a neighborhood of the closed unit disk and Φ(D) intersects the unit circle, then the operator 2 Φ(D) is hypercyclic on E2(ez). Note that, by [8, Proposition 1.4(a)], every Φ(D)-hypercyclic function for the space E2(ez) is a Φ(D)-hypercyclic entire function of exponential type ≤1 for the space H(C). Inspired by the facts about growth described so far, we investigate in this note the growth of hypercyclic entire functions with respect to certain sequences of infinite order linear differential operators and, in particular, with respect to a single operator Φ(D) satisfying suitable conditions. In a certain important particular case, the critical exponential growth type for hypercyclic entire functions is derived as a consequence. 2 Exponential type of hypercyclic functions Before establishing our results, we need some notation and several elementary facts. The multiplicity of an entire function Φ for the zero at the origin will be denoted by m(Φ) (∈N0). We can associate to each entire function Φ(z) = P∞ j=0 ajzjthe new entire function Φ∗(z) = P∞ j=0 |aj|zj. If |Φ(0)|<1, let us denote c(Φ) = min{|z|:|Φ(z)|= 1}, that is, c(Φ) is the least distance from the origin up to the “level curve” |Φ(z)|= 1. Note that c(Φ) ∈(0,∞). By continuity, |Φ(z)|<1 for all z∈B(0, c(Φ)). It is obvious that c(Φ∗)≤c(Φ). Observe that c(Φ) is the unique c > 0 with Φ(c) = 1 whenever the Taylor coefficients of Φ at the origin are ≥0. We are now ready to state our theorems about sequences of operators. The first one is negative and the second one is positive. Corresponding corollaries can be extracted regarding the hypercyclicity of a single operator Φ(D). Theorem 2.1 Assume that (Φn)is a sequence of entire functions, each of them with finite exponential type. Let c > 0such that the sequence (Φ∗ n(c)) is bounded. Then there is no hypercyclic entire function ffor (Φn(D)) such that |f(z)|=O(exp (cr) √r)as |z|=r→ ∞. In particular, there is no hypercyclic function fwith τ(f)< c. Proof. It is evident that it suffices to show that, for every entire function fwith the growth property of the statement, the sequence ((Φn(D)f)(0)) is not dense in C. In turn, it suffices to show that the latter sequence is bounded. For each n∈Nwe have Φn(z) = ∞ X j=0 aj,nzj, 3 for suitable aj,n ∈C(j∈N). Then Φ∗ n(z) = ∞ X j=0 |aj,n|zj. Assume that (Φ∗ n(c)) is bounded. Suppose that fis an entire function such that there exists a positive constant K1with M(f, r)< K1·exp(cr)/√r (r > 0). By Cauchy’s inequalities, |f(j)(0)| ≤ j!K1·exp(cr) rj+(1/2) (j∈N0;r > 0). By Stirling’s formula, there exists a positive constant K2with j!≤K2·jj+(1/2) · e−j(j∈N). By choosing r=j/c, one obtains that |f(j)(0)| ≤ K·exp jexp (−j)·jj+(1/2) ·cj jj+(1/2) , where K=c1/2K1K2, that is, |f(j)(0)| ≤ K·cj(j∈N), so |(Φn(D)f)(0)|=| ∞ X j=0 aj,nf(j)(0)|≤|Φn(0)|·|f(0)|+K· ∞ X j=1 |aj,n|cj = Φ∗ n(0)|f(0)|+K·(Φ∗ n(c)−Φ∗ n(0)) ≤Φ∗ n(c)·(|f(0)|+K). Thus, the sequence ((Φn(D)f)(0)) is bounded, as required. Corollary 2.2 Assume that Φis an entire function with finite exponential type and |Φ(0)|<1. Then there is no Φ(D)–hypercyclic entire function ffor which |f(z)|=O(exp (c(Φ∗)r) √r)as |z|=r→ ∞. In particular, there is no Φ(D)–hypercyclic entire function fwith τ(f)< c(Φ∗). Proof. We are going to apply Theorem 2.1. Set c=c(Φ∗). It suffices to see that (Φ∗ n(c)) is bounded, where Φn:= Φ ···Φ (ntimes). We have from the triangle inequality and the Cauchy product rule for series that Φ∗ n(c)≤ (Φ∗(c))n= 1 for every n. The conclusion follows and the proof is finished. Theorem 2.3 Assume that (Φn)is a sequence of entire functions such that m(Φn)→ ∞ (n→ ∞), in such a way that every Φnhas finite exponential type. Let us set A={z∈C: the sequence ( 1 Φn(z)) is bounded}. 4 (1) If c≥0and Ahas at least one accumulation point in B(0, c), then for every d > c there exists a dense linear submanifold of H(C)consisting, except for zero, of (Φn(D))–hypercyclic functions fwith τ(f)< d. (2) If c > 0and A∩B(0, c)is infinite, then there exists a dense linear submanifold of H(C)consisting, except for zero, of (Φn(D))–hypercyclic functions fwith τ(f)≤c. Proof. Let us try to apply Theorem 1.1. Fix d>cand pick any t∈(c, d). Take X=Xtand Y=H(C), where Xt:= {f∈H(C) : ||f||t<∞and || ∞ X j=k ajzj||t→0 (k→ ∞)} and ||f||t:= sup k∈N0 sup r>0 sup |z|=r (| ∞ X j=k ajzj|e−tr) whenever f(z) = P∞ j=0 ajzj. Define Tn:X→Yby Tn= Φn(D)|X(n∈N). Note that all functions in Xhave exponential type less than d. Note also that every function exp(az) with |a|< t is in X, because if aj(j∈N0) are its Taylor coefficients then |P∞ j=kajzj| ≤ e|a|rfor all kand all zwith |z|=r. As in [12], it is easy to see that ||·||tis a norm on Xwhich makes Xa Banach space with a topology which is stronger than that of uniform convergence on compacta, and that the set X0:= {polynomials}is dense in X. Hence each Tnis a continuous linear mapping from Xinto Y. If we fix a polynomial P then we get that the sequence (TnP) is eventually zero due to the condition that m(Φn)→ ∞ (n→ ∞). Therefore, trivially, TnP→0 for every P∈X0. On the other hand, under the hypotheses of (1), there exists a set A1⊂A with at least one finite accumulation point such that |a|< t for all a∈A1. From the existence of an accumulation point, a combination of Hahn-Banach Theorem, Riesz Theorem and Analytic Continuation Principle (like in, for instance, [11, Section 5]) yields that the set Y0:= span {eaz :a∈A1}is dense in H(C). By linearity, it is enough to show that, given a∈A1, there is a sequence (fn)⊂Xwith Tnfn→eaz (n→ ∞) in H(C) such that (fnk) converges in Xfor some strictly increasing sequence (nk)⊂N. Note that, with the notation of Theorem 1.1, Sneaz would be fn. For this, we define fn(z) = eaz Φn(a)(n∈N). Observe that each fnbelongs to X. Then Tnfn= Φn(D)( eaz Φn(a)) = eaz →eaz (n→ ∞). Since a∈A1, there exists (nk)⊂Nand α∈Cwith 1/Φnk(a)→α, hence fnk(z)→αeaz as k→ ∞ in X. Consequently, Theorem 1.1 applies and 5 we obtain that there is a residual (so dense) set in Xconsisting of (Φn(D))– hypercyclic functions. Under the hypotheses of (2), we would take this time X=\ t>c Xtand Y=H(C). It is easy to see that Xis a Fr´echet space when it is endowed with the translation-invariant distance d(f, g) = ∞ X j=1 2−j||f−g||cn 1 + ||f−g||cn , where (cn) is any sequence strictly decreasing to c. We define again Tn:X→Y (n∈N) by Tn= Φn(D)|X. Observe that the topology of Xis stronger than the compact-open topology, and that the set X0:= {polynomials}is dense in X. This time exp(az)∈Xwhenever |a| ≤ c, and all functions in Xhave exponential type ≤c. As before, the set Y0:= {eaz :a∈A1}is dense in H(C), where A1:= A∩B(0, c) this time. From here on, the proof runs through the same steps as part (1) and Theorem 1.1 can be again applied to produce a dense set of (Φn(D))-hypercyclic functions in X. Finally, observe that we have in fact obtained for each increasing sequence (nk)⊂Nthe existence of a dense set in X of (Φnk(D))–hypercyclic functions, because the above argument can be applied to every subsequence (Φnk). Since Xand Yare metrizable separable topological vector spaces (the separability of Xis a consequence of the fact that the set of polynomials is dense in X) we have that the hypotheses of Theorem 1.2 are fulfilled for (Tn). Hence there is a linear submanifold M⊂Xsatisfying that every function f∈M\ {0} is hypercyclic for (Φn(D)); in addition, Mis dense in X. But Xis dense in H(C) for the topology of this last space, because all polynomials are in Xand the topology of Xis stronger than that of local uniform convergence. Thus, Mis also dense in H(C), and the proof is finished. Corollary 2.4 Assume that Φis a nonconstant entire function of finite exponential type with Φ(0) = 0. We have: (1) Given d > c(Φ), there is a dense linear submanifold in H(C)consisting, except for zero, of Φ(D)–hypercyclic functions fwith τ(f)< d. (2) If |Φ(z)|= 1 on an infinite set of points of the circle of center at the origin and radius c(Φ), then there exists a dense linear submanifold in H(C)consisting, except for zero, of Φ(D)–hypercyclic functions fwith τ(f)≤c(Φ). 6 Proof. Note that m(Φn) = mn → ∞ (n→ ∞), where m > 0 is the multiplicity of Φ for the zero at the origin and Φn= Φ ···Φ (n-fold). A simple application of the Maximum Modulus Principle shows that the set {z∈C: |Φ(z)|>1}(⊂ {z∈C: (1/Φn(z)) is bounded}) has an accumulation point in B(0, c(Φ)). Hence part (1) of Theorem 2.3 applies and one obtains part (1) of this corollary. Part (2) is in turn derived from part (2) of Theorem 2.3, by considering the same sequence (Φn). Nevertheless, it should be pointed out that the conclusion of part (1) of the latter corollary can be deduced from the result of Chan and Shapiro mentioned in Section 1, even without assuming Φ(0) = 0 (hence the translations τa= exp(aD) are included). Indeed, the simple substitution f(z)7→ f(dz) (d > 0) yields that if Φ(dD) intersects the unit disk (which is the same as d>c(Φ)) then Φ(D) is hypercyclic on E2(edz), so Φ(D) is hypercyclic on H(C) and has a hypercyclic entire function with τ(f)≤d. The existence of the dense linear submanifold is a consequence of the fact that E2(edz) is dense in H(C) by using a general property of hypercyclic operators, namely, if Tis hypercyclic on a locally convex space X, then there is a dense T-invariant linear submanifold Mof Xsuch that each vector in M\ {0}is hypercyclic, see [7]. As for the special case Φ(D) = D, part (2) of Corollary 2.4 can be used to get a result containing MacLane’s one in Section 1: There is a dense linear manifold in H(C) consisting, except for zero, of D–hypercyclic functions fwith τ(f)≤1 (so τ(f) = 1 by Corollary 2.2 or [12]). Indeed, A={|z| ≥ 1}in this case. Note that here Φ(B(0,1)) intersects the unit circle. With this hypothesis, Chan and Shapiro [8, p. 1447] proposed the question that whether Φ(D) is hypercyclic on E2(ez). An affirmative answer to this would yield a corresponding affirmative answer for the problem that whether Φ(D) has a hypercyclic entire function of exponential type ≤1. For instance, as far as we know, even the case Φ(z) = 1+z 2(see [8, p. 1447]) remains unsolved. The next corollary is a joint consequence of Corollary 2.2 and of either part (1) of Corollary 2.4 or Chan-Shapiro’s result. Corollary 2.5 If Φis a nonconstant entire function with finite exponential type such that |Φ(0)|<1and its Taylor coefficients at the origin are ≥0then inf {τ(f) : fis hypercyclic for Φ(D)}=c(Φ). For instance, inf {τ(f) : {Pn j=0 n j(−1)n−jf(z+j) : n∈N}is dense in H(C)}= log 2. Indeed, just apply the corollary on Φ(z) = exp z−1 and take into account that exp(D) is the 1–translation operator. To finish, we propose the following open question: For Φ as in Corollary 7 2.5, give the exact critical rate of growth for Φ(D)-hypercyclic functions, as in [12]. ACKNOWLEDGEMENT The authors are grateful to the referee for helpful comments and suggestions. References [1] M. P. Aldred and D. H. Armitage, Harmonic analogues of G.R. MacLane’s universal functions, J. London Math. 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