Rate of growth of hypercyclic and frequently hypercyclic functions for the Dunkl operator

arXiv:1508.07180v1 [math.FA] 28 Aug 2015 Rate of growth of hypercyclic and frequently hypercyclic functions for the Dunkl operator L. Bernal-Gonz´alez and A. Bonilla Abstract For the Dunkl operator Λα(α > −1/2) on the space of entire functions on the complex space C, the critical rate of growth for the integral means Mp(f, r) of their hypercyclic functions fis obtained. The rate of growth of the corresponding frequently hypercyclic functions is also analyzed. 1 Introduction. In this paper, we consider the vector space H(C) of all entire functions C→C, endowed with the topology of local uniform convergence. Under this topology, H(C) becomes an F-space, that is, a completely metrizable topological vector space. Recall that a (continuous and linear) operator Ton a (Hausdorff) topological vector space Xis said to be hypercyclic if there exists a vector x∈X, also called hypercyclic, whose orbit {Tnx:n∈N} is dense in X. We refer the reader to the books [6] and [26] for rather complete accounts and further information on hypercyclic operators. In 1952, MacLane [32] proved that the differentiation operator D:H(C)→ H(C), given by Df(z) = f′(z), is hypercyclic. Moreover, he showed that D-hypercyclic entire functions can be of exponential type 1. In 1984, Duyos-Ruiz [19] they cannot be of exponential type less than 1. An optimal result on the possible rates of growth for the differentiation operator was subsequently obtained by Grosse-Erdmann [22] and, independently, by Shkarin [37]. Specifically, they obtained that there is no D- hypercyclic entire function ffor which there exists a constant C > 0 satisfying |f(z)| ≤ Cer √rfor |z|=rlarge enough but, given a function ϕ:R+→R+with ϕ(r)→ ∞ as r→ ∞, there exists a D-hypercyclic entire function such that |f(z)| ≤ ϕ(r)er √rfor |z|=rlarge enough. 2010 Mathematics Subject Classification. Primary 47A16; Secondary 30D15, 47B37, 47B38. Key words and phrases. Frequently hypercyclic operator, frequently hypercyclic vector, Frequent Hypercyclicity Criterion, rate of growth, entire function, Dunkl operator. 1 The notion of frequent hypercyclicity that was recently introduced by Bayart and Grivaux [4], [5]. We recall that the lower density of a subset Aof Nis defined as dens (A) = lim inf N→∞ #{n∈A:n≤N} N, where # denotes the cardinality of a set. A vector x∈Xis called frequently hypercyclic for Tif, for every non-empty open subset Uof X, dens {n∈N:Tnx∈U}>0. The operator Tis called frequently hypercyclic if it possesses a frequently hypercyclic vector. The problem of determining possible rates of growth of frequently hypercyclic entire functions for differentiation operator was studied in [10], [11] and [17]. More generally, the problem of isolating the possible rates of growth of hypercyclic or frequently hypercyclic entire functions for convolution operators on H(C) (operators that conmute with the differentiation operator) has been considered in [8], [13] and [14]. Finally, the rates of growth of hypercyclic entire functions for weighted backward shifts Bwon H(C) –considered as a special sequence space– have been established in [25]. More classes of hypercyclic nonconvolution operators on H(C) –different from composition operators– have been analyzed by a number of authors, see e.g. [2, 21, 28, 29, 31, 33, 34, 35]. One remarkable example is the Dunkl operator Λα:H(C)→H(C) (α > −1 2), which is a differential-difference operator given by Λαf(z) = d dz f(z) + 2α+ 1 zf(z)−f(−z) 2. Note that for α=−1 2we get Λα=D. The operator Λαwas introduced in 1989 by Dunkl [18]. It is connected to the theory of sampling signals, and there are in the literature a lot of papers dealing with the Dunkl operator, see for instance [16] and the references given in it. Here we will concentrate on the dynamical aspects of Λα. The hypercyclicity of this operator was established by Betancor, Sifi and Trimeche in [9] (see also [30]and [15]). The purpose of this note is to study the rate of growth of hypercyclic and frequently hypercyclic entire functions for the Dunkl operator. In fact, the critical rate of growth for the p-integral means Mp(f, r) of their hypercyclic functions fis obtained. Moreover, we get permissible and non-permissible rates of growth for the means of the corresponding frequently hypercyclic functions. 2 2 Preliminaries and notation. As usual, throughout this paper constants C > 0 can take different values at different occurrences. We write an∼bnfor positive sequences (an) and (bn) if an/bnand bn/anare bounded. For an entire function fand 1 ≤p < ∞we consider the p-integral means Mp(f, r) = 1 2πZ2π 0|f(reit)|pdt1/p (r > 0) and M∞(f, r) = sup |z|=r|f(z)|(r > 0). These means have been considered by Blasco, Bonet, Bonilla and Grosse- Erdmann (see [10, 11]) in order to establish the possible rates of growth of hypercyclic or frequently hypercyclic entire functions for the operator D. Some of the techniques from [10, 11, 22] will be used and generalized in this paper in order to analyze the corresponding problem for Λα. With this aim, some properties of the Dunkl operators are needed. Let us define dn(α) = 2n[n 2]! Γ(α+ 1) Γ[n+ 1 2] + α+ 1for n≥0 and α > −1 2, and dn(α) = 0 when n < 0, where [x] denotes the integer part of x. Notice that dn(α) tends rapidly to +∞as n→ ∞. Then, for every k∈N, we have (see e.g. [9, pag. 106]) that Λk α(zn) = dn(α) dn−k(α)zn−kfor all z∈C. Moreover, given f(z) = P∞ n=0 f(n)(0) n!zn∈H(C), the following holds: Λn αf(0) = f(n)(0) n!dn(α). By using Stirling’s asymptotic formula (see e.g. [1]) Γ(x) = (2π)1/2xx−1 2e−xΨ(x) for x > 0 (where Ψ(x)→1 as x→+∞) and the facts Γ(n+ 1) = n!, [n 2] + [n+1 2] = n(n≥0), one can easily obtain the following lemma, which will be used in the forthcoming sections. Lemma 1. For each α > −1/2, we have the equivalence dn(α)∼(n+α+ 1)n+α+1 en+α+1 . An important tool in the setting of linear dynamics is the Universality Criterion. There are several versions of it, see for instance [23] or [26]. The one given in Theorem 1 is sufficient for our goals. Prior to this, it is easy to extend the notion of hypercyclicity. A sequence of continuous linear mappings 3 Tn:X→Y(n≥1) between two topological vector spaces X, Y is said to be universal whenever there is a vector x0∈X–called universal for (Tn)– such that the set {Tnx0:n∈N}is dense in Y. Of course, an operator T:X→X is hypercyclic if and only if the sequence (Tn) of its iterates is universal. Theorem 1. Assume that Xand Yare topological vector spaces, such that X is a Baire space and Yis separable and metrizable. Let Tn:X→Y(n≥1) be a sequence of continuous linear mappings. Suppose also that there subsets X0⊂Xand Y0⊂Ythat are respectively dense in Xand Ysatisfying that, for any pair (x0, y0)∈X0×Y0, there are sequences {n1< n2<···< nk< ···} ⊂ Nand (xk)⊂Xsuch that xk→0, Tnkx0→0and Tnkxk→y0as n→ ∞. Then (Tn)is universal. In fact, the set of universal vectors for (Tn)is residual (hence dense)in X. Corresponding notion and criterium for frequent universal sequences will be given in Section 4. In that section the well-known Hausdorff–Young inequality (see for instance [27] or [36]) will be used several times. We state a version of it in the next theorem for the sake of completeness. Recall that if p∈[1,∞] then the conjugate exponent of pis the unique q∈[1,∞] satisfying 1 p+1 q= 1. Theorem 2. Consider the spaces Lp([0,2π]) of Lebesgue integrable functions [0,2π]→Cof order p. For each F∈Lp([0,2π]), let ˆ F(n) = 1 2πR2π 0F(t)e−int dt (n≥0) be the sequence of its Fourier coefficients. If 1< p ≤2and qis the conjugate exponent of pthen ∞ X n=0 |ˆ F(n)|q1/q ≤1 2πZ2π 0|F(t)|pdt1/p. 3 Hypercyclic entire functions for the Dunkl operator. For hypercyclicity with respect to the Dunkl operator, the rate of growth er/rα+1 turns out to be critical. We denote R+:= (0,+∞) and N0:= N∪{0}. Theorem 3. Let 1≤p≤ ∞. Then the following holds: (a) For any function ϕ:R+→R+with ϕ(r)→ ∞ as r→ ∞ there is a Λα-hypercyclic entire function fwith Mp(f, r)≤ϕ(r)er rα+1 for r > 0sufficiently large. (b) There is no Λα-hypercyclic entire function fsatisfying, for some C > 0, that Mp(f, r)≤Cer rα+1 for all r > 0. 4 Proof. Thanks to the elementary inequality Mp(f, r)≤Mq(f, r) for all r > 0 (1 ≤p≤q) it is enough to prove part (a) for the case p=∞and part (b) for the case p= 1. Let us prove (a). We can assume without loss of generality that ϕ:R+→ R+with ϕ(r)→ ∞ as r→ ∞ is monotonic and continuous on [0,+∞) with ϕ(0) >0. Consider the following space e X=f∈H(C) : sup r>0 M∞(f, r)rα+1 ϕ(r)er<∞. It is not difficult to see that e Xis a Banach space under the norm kfk:= supr>0 M∞(f,r)rα+1 ϕ(r)erthat is continuously embedded in H(C). In addition, we define the set Xas the closure of the polynomials in e X, and the set X0=Y0⊂H(C) as the collection of polynomials in z. Note that, trivially, X0is dense in Xand Y0is dense in Y:= H(C). Now, for n∈N:= {1,2,3,...}, we consider the mappings Tn:X→Y, Tn= Λn α|X, which are continuous, and Sn:Y0→X, Sn=Snwith S(zk) = dk(α) dk+1(α)zk+1, extended linearly to Y0. Now, fix a pair of polynomials P, Q, that is, (P, Q)∈X0×Y0. Then P(z) = Pp k=0 akzkand Q(z) = Pq k=0 bkzkfor certain a0,...,ap, b0,...,bq∈C. Take as (nk) the full sequence {1,2,3,...}of natural numbers and fn(z) := SnQ (with Sn=S◦S◦···◦S,ntimes) for all n∈N. It follows from the properties of Λαgiven in Section 2 that Tnfn=Q(so, trivially, Tnfn→Qas n→ ∞) and TnP→0 in H(C) as n→ ∞ (because Λn α(zk) = 0 if n≥k). Thus the conditions of the Universality Criterion (Theorem 1) are satisfied if we can show that {fn}n≥1converges to 0 in X. By linearity, one can assume that Q(z) = zk, a monomial (k∈N0), in which case lim n→∞SnQ(z) = lim n→∞ dk(α) dk+n(α)zk+n. Therefore, all we need to show is that lim n→∞ zn dn(α) converges to 0 in X. 5 To this end, fix ε > 0 as well as an n∈N. We choose R > 0 such that ϕ(r)≥1/ε for r≥R. Then we have that sup r≤R rα+1 ϕ(r)er rn dn(α)≤Rα+1 infr>0ϕ(r) Rn dn(α)→0 as n→ ∞. Moreover a simple calculation involving the derivative of rn+α+1e−rshows that sup r≥R 1 ϕ(r) rα+1 er rn dn(α)≤ε(n+α+ 1)n+α+1 en+α+1 dn(α), and Lemma 1 implies that the last term is bounded by Cε for any n∈N. This shows that limn→∞ zn dn(α)converges to 0 in X. In order to prove part (b), the use of the Cauchy estimates leads us to |Λn αf(0)|= f(n)(0) n!dn(α)≤dn(α) rnM1(f, r). Assume, by way of contradiction, that there is C > 0 such that M1(f, r)≤ Cer rα+1 for all r > 0. Hence we find that |Λn αf(0)| ≤ Cdn(α) rn+α+1 er≤Cdn(α) (n+α+ 1)n+α+1 en+α+1 for all n≥1. Now Lemma 1 implies that the sequence {Λn αf(0)}n≥1is bounded, so that f cannot be hypercyclic for Λα. This is the desired contradiction. Remark 1. An operator T:H(C)→H(C) is called a weighted backward shift if there is a sequence {an}n≥1⊂C\{0}such that (T f)(z) = ∞ X n=0 an+1cn+1zn(z∈C) provided that f(z) = P∞ n=0 cnzn. We denote this operator by T=B(an). It is not difficult to see that B(an)is well defined and continuous if and only if supn∈N|an|1/n <∞. In [24] (see also [7]) is was proved that, under the assumption supn∈N|an|1/n <∞, the operator B(an)is hypercyclic if and only if lim supn→∞ Qn k=1 ak 1/n =∞. This result contains MacLane’s theorem as a special case because the differentiation operator Dis the weighted backward shift B(an)with an=n(n∈N). Now, observe that, for α > −1/2, the Dunkl operator is B(an)with an=dn(α) dn−1(α). In this case, from Lemma 1 one easily derives that supn∈N|an|1 n<∞and lim supn→∞ Qn k=1 ak 1/n = lim supn→∞ dn(α)1 n=Climn→∞(n+α+1)n+α+1 n=∞. Then the Dunkl operator is hypercyclic (note that we get an alternative proof of this result given in [9]; as a matter of fact, a much more general result is shown in [9, Theorem 4.1]). 6 Moreover, Grosse-Erdmann proved in [24, Theorems 1–2] that, under the assumption that (|an|) is nondecreasing, the critical rate of growth of M∞(f, ·) allowed for a B(an)-hypercyclic entire function fis µ(r) = maxn≥0|rn/Qn k=1 ak|. If B(an)= Λαthen (|an|) is nondecreasing and µ(r) = maxn≥0|rn/dn(α)|, so the cited theorems of [24] are at our disposal. Nevertheless, we have opted for a more direct proof, which in turn yields critical rates for all Mp(f, ·) (1 ≤p≤ ∞). 4 Frequently hypercyclic entire functions for the Dunkl operator. First of all, it is easy to extend the notion of frequent hypercyclicity to a sequence of mappings, see [12]. Definition 1. Let Xand Ybe topological spaces, and let Tn:X→Y(n∈N) be a sequence of continuous linear mappings. Then an element x∈Xis called frequently universal for the sequence (Tn) if, for every non-empty open subset Uof Y, dens{n∈N:Tnx∈U}>0. The sequence (Tn) is called frequently universal if it possesses a frequently universal element. Of course, an operator T:X→Xon a topological vector space Xis frequently hypercyclic if and only if the sequences of its iterates (Tn) is frequently universal. In [12], a Frequent Universality Criterion was obtained that generalizes the Frequent Hypercyclicity Criterion of Bayart and Grivaux [5]. We state it here (see Theorem 4 below) only for Fr´echet spaces. Recall that a collection of series P∞ k=1 xk,j (j∈I) in a Fr´echet space Xis said to be unconditionally convergent, uniformly in j∈I,if for every continuous seminorm pon Xand every ε > 0 there is some N≥1 such that for every finite set F⊂Nwith F∩{1,2,...,N}=∅and every j∈Iwe have that p(Pk∈Fxk,j)< ε. Theorem 4. Let Xbe a Fr´echet space, Ya separable Fr´echet space, and Tn: X→Y(n∈N)a sequence of operators. Suppose that there are a dense subset Y0of Yand mappings Sn:Y0→X(n∈N)such that, for all y∈Y0, (i) Pk n=1 TkSk−nyconverges unconditionally in Y, uniformly in k∈N, (ii) P∞ n=1 TkSk+nyconverges unconditionally in Y, uniformly in k∈N, (iii) P∞ n=1 Snyconverges unconditionally in X, (iv) TnSny→yas n→ ∞. Then the sequence (Tn)is frequently universal. We note that the sums in (i) can be understood as infinite series by adding zero terms. As an auxiliary result we shall need the following estimate. In the following, we shall adopt the convention 1 2p= 0 for p=∞. 7 Lemma 2. Let 1≤q≤2and pbe the conjugate exponent of q. Then there is some C > 0such that, for all r > 0, we have ∞ X n=0 rqn (dn(α))q≤Cer rα+1 2+1 2pq . Proof. Consider the entire functions Eα(z;θ, β) = ∞ X n=0 zn (n+θ)βΓ(αn + 1) for α, θ > 0, β ∈R. In the special case of β= 0 these are the Mittag-Leffler functions, see for instance [20, Section 18.1]. Barnes [3, pp. 289–292] studied the functions Eα(z;θ, β) and, by using Stirling’s formula, he derived asymptotical expansions from which one deduces that, for 0 < α ≤2, (1) Eα(r;θ, β) = αβ−1r−β/αer1/α 1 + O(r−1/α)as r→+∞. Now, by the definition of dn(α), we get ∞ X n=0 rqn (dn(α))q≤∞ X n=0 (Γ(α+ 1))qr2qn (22nn! Γ(n+α+ 1))q+∞ X n=0 (Γ(α+ 1))qr(2n+1)q (22n+2 n! Γ(n+α+ 2))q = (Γ(α+1))q∞ X n=0 r2qn 22qn n!q(Γ(n+α+ 1))q+rq 22q ∞ X n=0 r2nq (22qn n!q(Γ(n+α+ 2))q Thanks to (1), we obtain for r > 0 that ∞ X n=0 r2qn 22qn n!q(Γ(n+α+ 1))q≤C∞ X n=0 ((qr)2q)n (n+ 1)q(α+1)−1/2Γ(2qn + 1) ≤C(qr)2q−q(α+1)−1 2 2qe(qr)2q1 2q ≤Cer rα+1 2+1 2pq , where the constant Cis not necessarily the same in each occurrence. Analogously, we have for all r > 0 that rq 22q ∞ X n=0 r2nq (22qn n!q(Γ(n+α+ 2))q≤C rqer rα+1+ 1 2+1 2pq ≤Cer rα+1 2+1 2pq . Finally, it is enough to add up both inequalities to get the desired result. Our first main result in this section gives growth rates for which Λα-frequently hypercyclic functions exist. 8 Theorem 5. Let 1≤p≤ ∞, and put a=α+1 2+1 2 max{2,p}. Then, for any function ϕ:R+→R+with ϕ(r)→ ∞ as r→ ∞, there is an entire function fwith Mp(f, r)≤ϕ(r)er rafor r > 0sufficiently large that is frequently hypercyclic for the Dunkl operator. Proof. Since Mp(f, r)≤M2(f, r) for 1 ≤p < 2, we need only prove the result for p≥2. Thus let 2 ≤p≤ ∞. We shall make use of the Frequent Universality Criterion (Theorem 4). Assuming without loss of generality that infr>0ϕ(r)> 0, we consider the vector space X:= nf∈H(C) : sup r>0 Mp(f, r)rα+1 2+1 2p ϕ(r)er<∞o. If we endow Xwith the norm kfk:= supr>0 Mp(f,r)rα+1 2+1 2p ϕ(r)erthen it is not difficult to see that (X, k·k) is a Banach space that is continuously embedded in H(C). Let Y0⊂H(C) be the set of polynomials, and we consider the mappings Tn= Λn α|X:X→H(C), which are continuous, and Sn:Y0→X, Sn=Snwith S(zk) = dk(α) dk+1(α)zk+1. Then we have for any polynomial fand any k∈Nthat k X n=1 TkSk−nf= k X n=1 Λn αf, that converges unconditionally convergent in H(C), uniformly for k∈N, because P∞ n=1 Λn αfis a finite series. Moreover, we have according to the properties given in Section 2 that TnSnf=ffor any n∈N, and ∞ X n=1 TkSk+nf=∞ X n=1 Snf. 9