Interpolation by hypercyclic functions for differential operators

Interpolation by hypercyclic functions for differential operators by L. Bernal-Gonz´alez∗ Affiliation and address for manuscript correspondence: Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad de Sevilla, Apdo. 1160, Avda. Reina Mercedes, 41080 Sevilla, Spain. E-mail: lb[email protected]. Phone: +34 954557118. Fax: +34 964557972. 2000 Mathematics Subject Classification: Primary 47A16. Secondary 30E05, 30E10, 47B38. ∗The author has been partially supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127, and by MEC Grants MTM2006-13997-C02-01 and Acci´on Especial MTM2006-26627-E. 1 Abstract We prove that, given a sequence of points in a complex domain Ω without accumulation points, there are functions having prescribed values at the points of the sequence and, simultaneously, having dense orbit in the space of holomorphic functions on Ω. The orbit is taken with respect to any fixed non-scalar differential operator generated by an entire function of subexponential type, thereby extending a recent result about MacLane-hypercyclicity due to Costakis, Vlachou and Niess. Key words and phrases: Hypercyclic function, differential operator, interpolation, mixing sequence of mappings. 2 1 Introduction In 1929, Birkhoff [6] proved the existence of an entire function fwhich is universal in the sense that the sequence {f(·+an) : n≥1}of its translates (a∈C\ {0}) is dense in the space H(C) of entire functions. Later, in 1952, MacLane [15] demonstrated the same density property for the sequence {f(n):n≥1}of derivatives of some entire function f. Formally speaking, the translation operator τa:f7→ f(·+a) and the derivation operator D: f7→ f0acting on the space H(C) are hypercyclic, see Section 2. In 1991, Godefroy and Shapiro [10] unified both above results by establishing the hypercyclicity on H(C) of any nonscalar differential operator Φ(D), generated by an entire function Φ with exponential type. Birkhoff’s theorem and MacLane’s theorem correspond, respectively, to the cases Φ(z) = exp(az), Φ(z) = z. In a different direction, Weierstrass’ interpolation theorem (see [18]) asserts the existence of holomorphic functions on a domain Ω of the complex plane Chaving prescribed values at a sequence of points without accumulation points. Recently, Costakis and Vlachou [9] and, independently, Niess ([16],[17]) have carried out the interesting task of combining both properties of hypercyclicity and interpolation. To be more specific, they have proved the existence of a holomorphic function f(in fact, of a plethora of them) that is MacLane-universal and, simultaneously, takes prescribed values at the points of a given set without accumulation points in Ω, whenever Ω is simply connected (Ω = Cin [16],[17]). They have also established a number of results concerning interpolation for functions enjoying either Birkhoff universality on C([9],[17]), or multiplicative universality in C\ {0}[17], or universality in a domain with respect to some similarities z7→ az +b[9], or universality (on multiply connected domains) of their Taylor partial sums 3 [9]. In [16], [17], the universality-interpolation properties are even combined with prescribed zeros with given multiplicities. It seems that Costakis [8] was the first analyst that dealt with the problem of existence of interpolating universal functions, precisely in the setting of universal Taylor series on simply connected domains. Our aim in this paper is to extend the former result by Costakis–Vlachou and Niess. Namely, we generalize their assertion to differential operators generated by entire functions Φ with subexponential type. Section 2 is devoted to give a number of pertinent definitions and instrumental results, while Section 3 contains the exact statement of our theorem, together with its proof and a result about linear hypercyclic structure in a special case. 2 Terminology and preliminary results Let Ω be a domain in C, that is, Ω is a nonempty connected open subset of C. The vector space H(Ω) of holomorphic functions on Ω becomes a complete metrizable separable space when endowed with the topology of uniform convergence on compacta. In particular, it is a second-countable Baire space. A domain Ω is said to be simply connected provided that its complement with respect to the extended complex plane is connected. If S⊂C, then S0denotes the interior of S. Let Φ be an entire function. Then Φ is said to be of exponential type if there are constant A, B ∈(0,+∞) such that |Φ(z)| ≤ Aexp(B|z|) (z∈C). And Φ is called of subexponential type whenever, given ε > 0, there is a constant A=A(ε)∈(0,+∞) such that |Φ(z)| ≤ Aexp(ε|z|) (z∈C). In other words, Φ is of exponential (subexponential, resp.) type if and only if it has either growth order <1 or growth order = 1 and type <+∞ 4 (either growth order <1 or growth order = 1 and type 0, resp.). If Ω is a domain and Φ(z) := P∞ n=0 cnznis an entire function of subexponential type, then the expression Φ(D) := P∞ n=0 cnDndefines a (linear, continuous) operator Φ(D) : f∈H(Ω) 7→ P∞ n=0 cnf(n)∈H(Ω), see [1]. Note that Φ(D) is a (generally, infinite order) linear differential operator with constant coefficients. If Ω = C, then the operator Φ(D) is well-defined even if Φ is of exponential type. In fact, an operator Ton H(C) has the form T= Φ(D) with Φ of exponential type if and only if Tcommutes will all translations τa (a∈C), see [10]. For the following concepts and results on universality and other dynamical properties, the reader is referred to [14], [12], [13], [7] and [11]. Assume that Tn:X→Y(n∈N:= {1,2, . . . }) is a sequence of continuous mappings between two topological spaces X, Y . Then {Tn}n≥1is said to be: (a) universal if there exists a point x0∈X(called universal for {Tn}n≥1) whose orbit {Tnx0:n∈N}is dense in Y. (b) topologically transitive if, given nonempty open subsets U⊂X, V ⊂Y, there is N∈Nsuch that TN(U)∩V6=∅. (c) topologically mixing if, given nonempty open subsets U⊂X, V ⊂Y, there is N∈Nsuch that Tn(U)∩V6=∅for n≥N. If T:X→Xis a continuous selfmapping, then Tis called universal (topologically transitive, topologically mixing, resp.) provided that the sequence {Tn}n≥1of its iterates is universal (topologically transitive, topologically mixing, resp.). If X, Y are topological vector spaces and the Tn(or T, if we are dealing with selfmappings) are linear, then it is customary to say hypercyclic instead of universal. We denote U({Tn}n≥1) = {x∈X:xis 5 universal for {Tn}n≥1}and, if X=Y,U(T) = {x∈X:xis universal for T}=U({Tn}n≥1). Recall that a subset Aof a Baire topological space Xis residual when it contains a dense Gδsubset. It can be said that a residual subset is “topologically large” in X. The connection between universality and transitivity is given by the following well-known result. Birkhoff’s transitivity theorem. Assume that Tn:X→Y(n∈N) is a sequence of continuous mappings between two topological spaces X, Y . Suppose that Xis Baire and that Yis second-countable. Then the following are equivalent: (i) {Tn}n≥1is topologically transitive. (ii) {Tn}n≥1is densely universal, that is, U({Tn}n≥1)is dense in X. (iii) U({Tn}n≥1)is residual in X. One easily derives that, under the same hypotheses on Xand Y, the sequence {Tn}n≥1is topologically mixing if and only if {Tnk}k≥1is densely universal for every subsequence {n1< n2<· · · } ⊂ N. Godefroy and Shapiro [10] proved the hypercyclicity of any operator Φ(D) on H(C), with Φ a nonconstant entire function of exponential type. In fact, they worked in CN, and their result can be extended to a Runge domain Ω in CN(if N= 1, Runge equals simply connected) whenever Φ is of subexponential type (see [2] and [5], where sequences of operators Φn(D) are also considered). The approach followed by Godefroy and Shapiro shows that, in fact, the operators Φ(D) are mixing. In Section 3, we will need the next four lemmas. Lemma 2.1 is an analogue to an auxiliary bounded interpolation result given in [9], except that in ours exponentials are employed instead of polynomials. Lemma 2.2 fol- 6 lows from a well-known denseness assertion whose proof can be found in [10]. Lemma 2.3 says that the differential operators induced by entire functions of subexponential type are “internally controlled”. Its easy proof is sketched in [4]. Finally, Lemma 2.4 furnishes a statement about the strong linear structure of a topologically mixing sequence of linear mappings. By ea(a∈C) we denote the function ea(z) := exp(az), while spanX0will stand for the linear span of a subset X0of a vector space. Lemma 2.1. Assume that Lis a compact subset of C, that a1, . . . , amare different points in L, and that Ais a nonempty open subset of C. Then there are a constant M=M(L, a1, . . . , am, A)∈(0,+∞)and a finite set of functions {α1, . . . , αm} ⊂ span{ea:a∈A}, depending only on Aand the points a1, . . . , am, satisfying the following property: Given a pair f, h of complex-valued functions defined on L, there exists a function F:L→C satisfying: (a) supz∈L|F(z)−h(z)| ≤ Msupz∈L|f(z)−h(z)|. (b) F(aj) = h(aj) (j= 1, . . . , m). (c) F−f=Pm j=1(h(aj)−f(aj))αj. Proof. We can assume that A6=C. Let us choose a point c∈Aand select a positive number dsatisfying d < 1 2minf{|z−c|:z∈C\A}(1) and d < min j,l∈{1,...,m} j6=l 1 |aj−al|.(2) We define Πj(z) := Y l∈{1,...,m}\{j} (ed(z−al)−1) (j= 1, . . . , m). 7 From (2), it follows that 0< d|aj−al|<1<2π for all j, l ∈ {1, . . . , m}with j6=l, so Πj(aj)6= 0 for all j∈ {1, . . . , m}. In addition, Πj(al) = 0 whenever j6=l. Now, we set αj(z) := ec(z−aj)Πj(z) Πj(aj)(j= 1, . . . , m) and M:= 1 + m X j=1 sup z∈L |αj(z)|. Observe that each function αjis a finite linear combination of functions of the form ec+nd (0 ≤n≤m). But each point c+nd is in Abecause of (1). Hence the functions αjare in span{ea:a∈A}. Finally, if we fix functions f, h :L→Cand define F:= f+Pm j=1(h(aj)− f(aj))αj, then it is straightforward that (a), (b) and (c) are fulfilled. Lemma 2.2. Let A⊂Cbe a subset with some finite accumulation point, and Ω⊂Cbe a simply connected domain. Then the set span{ea:a∈A}is dense in the space H(Ω). In particular, this holds if Ais a nonempty open subset of C. Lemma 2.3. Let Ω⊂Cbe a domain and Φbe an entire function of subexponential type. Assume that K, L are compact sets in Ωwith L⊂K0. Then there exists a constant C=C(K, L)∈(0,+∞)such that sup z∈L |(Φ(D)f)(z)| ≤ Csup z∈K |f(z)|for all f∈H(Ω). 8 Lemma 2.4. Let X, Y be two metrizable separable topological vector spaces, such that Xis Baire. If Tn:X→Y(n∈N)is a topologically mixing sequence of continuous linear mappings, then there exists a dense linear manifold D ⊂ Xsuch that D \ {0}⊂U({Tn}n≥1). Proof. Note that Xis Baire, Yis second-countable and, by Birkhoff’s transitivity theorem, each sequence {Tnk}k≥1with {n1< n2<· · · } ⊂ Nis densely hypercyclic. Then the result follows from [3, Theorem 2]. 3Φ(D)-hypercyclicity with interpolation In this section, we establish our main result. For this, a bit of additional notation is needed. Let Ω ⊂Cbe a domain. By CNwe denote, as usual, the class of sequences of complex numbers, while ω(Ω) will stand for the set of sequences of pairwise different points in Ω without accumulation points in Ω. If {an}n≥1∈ω(Ω) and {bn}n≥1∈CN, then we denote (as in [9]) Γ = Γ({an}n≥1,{bn}n≥1) := {f∈H(Ω) : f(an) = bnfor all n∈N}. Observe that Γ is nonempty (by Weierstrass’ interpolation theorem). In addition, Γ is closed in H(Ω), hence completely metrizable, so a Baire space. Costakis and Vlachou [9] have proved that, if Ω is simply connected, the set U(D)∩Γ is residual in Γ. This admits an extension as follows. Theorem 3.1. Let Ω⊂Cbe a simply connected domain and Φbe a nonconstant entire function of subexponential type. Assume that {an}n≥1∈ω(Ω) and {bn}n≥1∈CN. Consider the corresponding set Γ = Γ({an}n≥1,{bn}n≥1). Then the sequence of mappings Tn:= Φ(D)n|Γ: Γ −→ H(Ω) (n≥1) 9 3.1, the sequence Tn: Γ0→H(Ω) (n∈N) defined as Tn= Φ(D)n|Γ0is topologically mixing. 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