Powers of Hypercyclic Functions for Some Classical Hypercyclic ... - UPV

Integr. equ. oper. theory Online First c 2007 Birkh¨
auser Verlag Basel/Switzerland DOI 10.1007/s00020-007-1490-4

Integral Equations and Operator Theory

Powers of Hypercyclic Functions for Some Classical Hypercyclic Operators R. M. Aron, J. A. Conejero, A. Peris and J. B. Seoane–Sep´ ulveda Abstract. We show that no power of any entire function is hypercyclic for Birkhoff’s translation operator on H(C). On the other hand, we see that the set of functions whose powers are all hypercyclic for MacLane’s differentiation operator is a Gδ -dense subset of H(C). Mathematics Subject Classification (2000). Primary 47A16; Secondary 30D15. Keywords. Hypercyclic vectors, universal functions.

1. Introduction and preliminaries Let X be a separable, infinite-dimensional F -space. A linear and continuous operator T defined on X is said to be hypercyclic if there exists x ∈ X such that its orbit under T , {T n x : n ∈ N}, is dense in X. The first two examples of hypercyclic operators were given in the space H(C) of entire functions endowed with the compact-open topology. In 1929, Birkhoff saw that the translation operator is hypercyclic on H(C) [3]. Later, MacLane proved that the derivative operator is also hypercyclic [16]. In both cases, the authors provide the construction of a hypercyclic function. A revised proof of both results can be found in [1] (see also [10].) For further information about the construction and properties of these hypercyclic entire functions see [17, 4, 5, 9, 12, 7, 15, 2, 6] The Baire Category Theorem provides a Gδ -dense set of hypercyclic vectors for both of them, see [13, 14] for a exhaustive survey of results concerning hypercyclicity of operators. Besides, they also share a common dense manifold of hypercyclic vectors (see the proof of Theorem 5.1 in [11]). However, nothing more is known concerning the structure of the set of hypercyclic vectors for these operators. The second author was supported in part by GVA, grant CTESPP/2005. The second and third authors were supported in part by MEC and FEDER, Project MTM200402262 and Research Net MTM2006-26627-E.

2

Aron, Conejero, Peris and Seoane-Sep´ ulveda

IEOT

Our purpose in this note is to study the behaviour of the powers of the hypercyclic vectors of both the Birkhoff translation and the MacLane differentiation operators. Despite the fact that these operators share many properties, we will see that the results obtained for powers are completely different.

2. Birkhoff ’s operator In [3], Birkhoff constructed a universal entire function, f , for the operator τ1

: H(C) f (z)

−→ →

H(C) f (z + 1)

One could wonder what kind of structure the set HC(τ1 ) = {f ∈ H(C) : f is hypercyclic for τ1 } has. As we have mentioned this is a Gδ -dense set. Here we see that, if we denote Bk := {f ∈ H(C) : f k ∈ HC(τ1 )}, k ∈ N, then Bk = ∅ for k > 1, in particular if f ∈ HC(τ1 ), then no power of f can be hypercyclic for τ1 . In order to do this, we will use a well known result by Hurwitz related to zeros of limits of entire functions (see, e.g. [8, p. 152]). Our main theorem in this section characterizes the closure of orbits of the powers of the hypercyclic functions for Birkhoff’s translation operator: Theorem 2.1. Let 1 < p ∈ N, f ∈ HC(τ1 ), and g ∈ H(C). If the order of each zero of g is a multiple of p, then g ∈ Orb(τ1 , f p ). Proof. Suppose that the order of each zero of g is a multiple of p. Let us call (an )n the sequence of non-zero zeros of g. Each an has multiplicity pmn with mn ∈ N, n ∈ N, and 0 has multiplicity pm, for some m ∈ N ∪ {0}. By Weierstrass’s theorem [8, Ch. VII, Th. 5.13], there is a sequence (pn )n of integers, and an entire function ϕ, such that g(z) = z pm eϕ(z)

∞
i=1

i Eppm (z/ai ), i

  where E0 (z) := 1 − z, and Eq (z) := (1 − z) exp z + z 2 /2 + . . . + z q /q , for q ≥ 1. Also, by Weierstrass’s theorem, the sequence (pn )n can be chosen in order to have ∞ that i=1 Epmi i (z/ai ) is also an entire function. Let us define m ϕ(z)/p

g˜(z) = z e

∞
i=1

Epmi i



z ai

 .

Next, since f ∈ HC(τ1 ), for any compact set K ⊂ C, there is a sequence (nj )j ∈ N with f (z + nj ) − g˜(z)K → 0,

Powers of Hypercyclic Functions

3

as j → ∞. It follows that f p (z + nj ) − g(z)K ≤ R(K, p) · f (z + nj ) − g˜(z)K → 0, as j → ∞, where R(K, p) > 0 is a constant that only depends on K and p, and
g ∈ Orb(τ1 , f p ). Theorem 2.2. Let 1 < p ∈ N, f, g ∈ H(C). If g ∈ Orb(τ1 , f p ), then the order of each zero of g is a multiple of p. Proof. Suppose that we consider a zero z0 of g. Take a closed disk D centered at z0 with no other zeros of g. By hypothesis there is a sequence (nj )j ⊂ N verifying f p (z + nj ) → g(z) as j → ∞, uniformly on D. By Hurwitz’s theorem, there is some n ∈ N such that the total number of zeros (counting multiplicity) of f p (z + n) and g(z) in D coincide. Therefore the order of z0 is a multiple of p.
From the previous theorems we have the following corollaries: Corollary 2.3. Let p, q ∈ N, p > 1, and f ∈ HC(τ1 ). Then z q ∈ Orb(τ1 , f p ) ⇐⇒

q ∈ N. p

Corollary 2.4. The set Bk := {f ∈ H(C) : f k ∈ HC(τ1 )} = ∅ for every k > 1. Clearly, all the previous results also hold for any general Birkhoff operator, namely τt (f )(z) = f (z + t), with t ∈ C \ {0}.

3. MacLane’s operator In [16], MacLane constructed a universal entire function for the differentiation operator D : H(C) −→ H(C) f (z) → f  (z) on H(C). Now we consider the following set, for k ∈ N, Mk := {f ∈ H(C) : f k ∈ HC(D)}. As we did in the previous section, one could ask what kind of structure the sets Mk have. In contrast to the results for τt , we have the following: Theorem 3.1. For every k ∈ N, Mk is a Gδ -dense set.

4

Aron, Conejero, Peris and Seoane-Sep´ ulveda

IEOT

Proof. Fix k ∈ N and let (Un )n be a countable basis of open sets in H(C). For every n ∈ N, we define the set Gn,k := {f ∈ H(C) : there exists j ∈ N such that Dj f k ∈ Un }. Fix n ∈ N. Clearly, Gn,k is open and non-void. We will show that it is also  m
i dense. Consider ε > 0, an arbitrary polynomial p(z) = i=0 ai z and q(z) =  m i i=0 bi z ∈ Un . Without loss of generality, bi = 0 for 0 ≤ i ≤ m. For any ε > 0 and any compact set K ⊂ C we will prove that there exist f (z) ∈ H(C) and j ∈ N f k (z) = q(z), and we will be done. such that for ||f (z) − p(z)||K < ε and Dj  m Let f (z) := p(z) + r(z) with r(z) := i=0 ci z i+n , for some n > m and the ci will be determined in order to obtain Dj f k (z) = q(z). Let j := (k − 1)m + kn. The derivative Dj f k (z) = Dj rk (z) is a polynomial of degree m, where the coefficient of z m−l , 0 ≤ l ≤ m, is  (km + kn − l)!  (m − l)!





(sm ,...,sm−l )∈Al

k sm sm−1 . . . sm−l

  sm−l  sm−1 , (3.1) . . . cm−l csmm cm−1

with

Al :=

(sm , . . . , sm−l ) ∈

Nl+1 0

m

:

i=m−l

si = k,

m

 isi = km − l ,

(3.2)

i=m−l

since the powers of z accompanying the ci have to add up to km − l, and   k sm sm−1 . . . sm−l is the corresponding multinomial coefficient. If we identify the coefficients of q(z) with the coefficients of Dj f k (z), we have a non-linear triangular system that can be easily solved if we begin with the coefficients of higher degree and go down. Besides, we have to show that if n is big enough, then the coefficients ci are small enough in order to have that f (z) is as close to p(z) as we want. Comparing the m-th coefficients gives that  1/k m!bm cm = . (3.3) (km + kn)! After identifying the (m − 1)-st coefficients of q(z) and Dj f k (z),we have n bm−1  1+ . (3.4) bm m Each of the coefficients, ci , depends on n, i.e. each ci can be seen as a sequence k = o(1/ (km + kn)!) and in n. Thus, from (3.3) and (3.4), we can say that c m  cm−1 = o(n/ k (km + kn)!). To conclude analogous statements for the rest of the cm−1 = cm

Powers of Hypercyclic Functions

5

coefficients we proceed as follows. Take 2 ≤ l ≤ m, and suppose that cm−i =  i k o(n / (km + kn)!) for 0 ≤ i ≤ l − 1, so that (km + kn − l)! sm sm−1 sm−l+1 cm cm−1 . . . cm−l+1 = o(1), (m − l)! for any choice (sm , sm−1 , . . . , sm−l+1 , 0) ∈ Al (see (3.2)). Thus, (km + kn − l)! k−1 cm cm−l = o(1), (3.5) (m − l)!  and then we obtain that cm−l = o(nl / k (km + kn)!). We should observe that, to solve the system, we need that all the ci in the previous process have to be non-zero. If it was not the case, a suitable modification of q(z) solves the problem. Finally, the set ∩∞ n=1 Gn,k is a second category set in H(C), which coincides with Mk , and this concludes the proof of the theorem.
As we have previously seen, the sets Mk (k ≥ 1) are Gδ -dense sets. From this fact it follows that ∩∞ k=1 Mk is a Gδ -dense set as well. To summarize, we can give the following result: Theorem 3.2. There exists f ∈ H(C) such that f k ∈ HC(D) for every k ∈ N. Moreover, this behaviour is generic, i.e. the following set is residual {f ∈ H(C) : f k ∈ HC(D) for every k ∈ N}.   It is also interesting to notice that B1 ∩ ∩∞ j=1 Mk is a Gδ -dense set as well. Acknowledgment We would like to thank Luis Bernal, Antonio Bonilla, and K. Grosse-Erdmann for pointing out to us a gap in the proof of Theorem 2.1 and for several discussions and helpful comments. We also want to thank the referee for helpful comments and remarks. The second author acknowledges the hospitality he received from the Department of Mathematical Sciences at Kent State University during NovemberDecember, 2005, while this paper was being written.

References [1] R. Aron and D. Markose, On universal functions. J. Korean Math. Soc. 41 (2004), 65–76. [2] L. Bernal-Gonz´ alez and A. Bonilla, Exponential type of hypercyclic entire functions. Archiv Math. 78 (2002), 283–290. [3] G.D. Birkhoff, D´emonstration d’un th´eor`eme ´el´ementaire sur les fonctions enti` eres. C. R. Acad. Sci. Paris 189 (1929), 473–475. [4] C. Blair and L.A. Rubel, A universal entire function. Amer. Math. Monthly 90 (1983), 331–332.

6

Aron, Conejero, Peris and Seoane-Sep´ ulveda

IEOT

[5] C. Blair and L.A. Rubel, A triply universal entire function. Enseign. Math. 30 (1984), 269–274. [6] A. Bonilla and K.G. Grosse-Erdmann, On a theorem of Godefroy and Shapiro. Int. Equat. Oper. Theory 56 (2006), 151-162. [7] K.C. Chan and J.H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40 (1991), 1421–1449. [8] J.B. Conway, Functions of One Complex Variable. Springer-Verlag, Berlin/New York, 1978. [9] S.M. Duyos Ruiz, Universal functions and the structure of the space of entire functions. Dokl. Akad. Nauk. SSSR 279 (1984), 792–795. [10] G. Fern´ andez and A.A. Hallack, Remarks on a result about hypercyclic nonconvolution operators. J. Math. Anal. Appl. 309 (2005), 52–55. [11] G. Godefroy and J.H. Shapiro, Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98 (1991), 229–269. [12] K.G. Grosse-Erdmann, On the universal functions of G.R. MacLane. Complex Variables Theory Appl. 15 (1990), 193–196. [13] K.G. Grosse-Erdmann, Universal families and hypercyclic operators. Bull. Amer. Math. Soc. 36 (1999), 345–381. [14] K.G. Grosse-Erdmann, Recent developments in hypercyclicity. Rev. R. Acad. Cien. Serie A Mat. 97 (2003), 273–286. [15] W. Luh, V.A. Martirosian, and J M¨ uller, Universal entire functions with gap power series. Indag. Math. (N.S.) 9 (1998), 529–536. [16] G.R. MacLane, Sequences of derivatives and normal families. J. Analyse Math. (1952), 72–87. [17] W. Seidel and J.L. Walsh, On approximation by euclidean and non-euclidean translations of an analytic function. Bull. Amer. Math. Soc. 47 (1941), 916–920. R. M. Aron Department of Mathematical Sciences, Kent State University, Kent, OH44242, USA e-mail: [email protected] J. A. Conejero Departament de Matem` atica Aplicada and IMPA-UPV, F. Inform` atica Universitat Polit`ecnica de Val`encia, E-46022 Val`encia, Spain e-mail: [email protected] A. Peris Departament de Matem` atica Aplicada and IMPA-UPV, E.T.S. Arquitectura Universitat Polit`ecnica de Val`encia, E-46022 Val`encia, Spain e-mail: [email protected] J. B. Seoane–Sep´ ulveda Facultad de Ciencias Matem´ aticas, Departamento de An´ alisis Matem´ atico Universidad Complutense de Madrid, Plaza de las Ciencias 3, E-28040 Madrid, Spain e-mail: [email protected] Submitted: April 19, 2006 Revised: December 7, 2006