Order of growth of distributional irregular entire functions for the differentiation operator

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1 Order of growth of distributional irregular entire functions for the differentiation operator Luis Bernal-González and Antonio Bonilla arxiv: v [math.cv] 8 Aug 05 June, 08 Abstract We study the rate of growth of entire functions that are distributionally irregular for the differentiation operator D. More specifically, given p [, ] and b (0,a), where a =, we prove that there exists a distributionally irregular entire function f for the operator D such that its p-integral max{,p} mean function M p (f,r) grows not more rapidly than e r r b. This completes related known results about the possible rates of growth of such means for D-hypercyclic entire functions. It is also obtained the existence of dense linear submanifolds of H(C) all whose nonzero vectors are D-distributionally irregular and present the same kind of growth. Introduction In 988, Beauzamy [5], when trying to describe the erratic dynamics of certain vectors under the action of concrete operators, introduced the following notion. Definition. Let X be a Banach space and T : X X be a bounded operator. A vector x 0 X is said to be irregular for T if liminf T n x = 0 and limsup T n x =. Inspired by this definition, and by the notion of distributional chaotic mapping due to Schweizer and Smítal ([5], see also [4]), in [6] it is considered the stronger property contained in Definition below. Recall that, if A is a subset of the set N of positive integers, then its upper density and its lower density are respectively defined by dens(a) = limsup card(a {,,...,n}), dens(a) = liminf n card(a {,,...,n}). n 00 Mathematics Subject Classification. Primary 30D5; Secondary 5A03, 30H50, 47A6, 47B38. Key words and phrases. Differentiation operator, irregular vector, distributionally irregular vector, hypercyclic operator, frequently hypercyclic operator, rate of growth, entire function.

2 Definition. Let X be a Banach space and T : X X be a bounded operator. A vector x 0 X is said to be distributionally irregular for T if there are increasing sequences of positive integers A = (n k ) k and B = (m k ) k such that dens(a) = = dens(b), lim k T n k x = 0 and lim k T m k x 0 =. Now, let Y be a Fréchet space, that is, Y is a vector space endowed with an increasing sequence ( k ) k N of seminorms (called a fundamental sequence of seminorms) that defines a metric d(x,y) := k= k min{, x y k} (x,y Y), under which Y is complete. By B(Y) we have denoted, as usual, the set of all continuous linear operators T : Y Y. The following concepts, that are a generalization of the above ones to Fréchet spaces, were introduced in []. Definition 3. Given T B(Y) and x 0 Y, we say that x 0 is an irregular vector for T if there are m N and strictly increasing sequences (n k ) and (j k ) of positive integers such that lim k Tn k x 0 = 0 and lim T j k x 0 m =. k Definition 4. Given T B(Y) and x 0 Y, we say that x 0 is a distributionally irregular vector for T if there are m N and A,B N with dens(a) = = dens(b) such that lim n A T n x 0 = 0 and lim T n x 0 m =. Let us consider the Fréchet space H(C) of entire functions endowed of the family of seminorms f,b(0,k) (k N). Here B(a,r) denotes, as usual, the open ball in the complex plane C with center a and radius r > 0; and, for a nonempty set A C, we have set f,a := sup{ f(z) : z A}. Hence the metric d(f,g) = min{, f k,b(0,k) } defines the topology of H(C). If D : f k= H(C) f H(C) is the differentiation operator, then the above definitions read as follows. Definition 5. Given f H(C), we say that f is an irregular function for D if there are m N and increasing sequences (n k ) and (j k ) of positive integers such that lim k Dn k f = 0 and lim D j k f k,b(0,m) =. Definition 6. Given f H(C), we say that f is a distributionally irregular function for D if there are m N and A,B N with dens(a) = dens(b) = such that lim n A n B D n f = 0 and lim D n f,b(0,m) =. n B

3 In [] is proved that D admits distributionally irregular entire functions. In fact, it is shown in [] that there are T-distributionally irregular entire functions for any given non-scalar operator T on H(C) that commutes with D. Another important property related to the dynamics of operators is that of hypercyclicity. If T is an operator defined on a topological vector space X and x 0 X, thenthevector x 0 iscalled hypercyclic fort provided that theorbit {T n x 0 : n N} isdenseinx; and,accordingtobayartandgrivaux[], x 0 iscalledfrequentlyhypercyclic for T whenever the following stronger property is satisfied: for any prescribed nonempty open set U X, dens({n N : U T n x 0 }) > 0. The existence of entire functions which are hypercyclic (frequently hypercyclic) for D is known since MacLane [3] (Bayart and Grivaux [], resp.). For excellent accounts of these topics, we refer the reader to the books [3, 0]. Notice that, if X is a Fréchet space, every T-hypercyclic vector is T-irregular. Many papers have been devoted to study the rate of growth of entire functions thatarehypercyclicorfrequentlyhypercyclicfor D,seeforinstance[9,3,4,5,6, 7, 8, 9, 6]. In this paper, we aim to study the rate of growth of entire functions that are irregular or distributionally irregular for the differentiation operator. The growth for the irregular case will be completely determined, while lower and upper bounds will be provided for the growth of D-distributionally irregular functions. This research is completed by analyzing the distributional irregularity in weighted Banach spaces as well as the existence of large vector subspaces of D-distributionally irregular functions satisfying the mentioned growth conditions. Order of growth of distributional irregular entire functions For every r > 0, every entire function f and p < we will consider the integral p-means ( π ) /p M p (f,r) = f(re it ) p dt π 0 as well as the function M (f,r) = sup z =r f(z). In 990, Grosse-Erdmann [9] discovered that there are not D-hypercyclic entire functions f with M (f,r) growing not more rapidly than e r / r, while there do exist D-hypercyclic entire functions growing under any prescribed rate speeder than e r / r. Our first result states that the critical order of growth for hypercyclic and irregular entire functions for the differentiation operator is the same, and that this is independent of the p-mean considered. Theorem 7. Let p. We have: (a) For any function ϕ : R + R + with ϕ(r) as r there is a D-irregular entire function f with M p (f,r) ϕ(r) er r for r > 0 sufficiently large. 3

4 (b) There is no D-irregular entire function f that satisfies where C is a positive constant. M p (f,r) C er r for r > 0, Proof. Part (a) follows from the corresponding result for D-hypercyclic functions [9] (see also [6]), since all hypercyclic vectors are irregular. As for (b), assume that f is an entire function satisfying M p (f,r) C er r (r > 0) for some C > 0. Fix m N and a radius R > m. From Cauchy s integral formula for derivatives of holomorphic functions, we get for all n N that D n f(z) = πi ξ =R f(ξ) dξ (z B(0,r)). (ξ z) n+ From here and the triangle inequality one derives that D n f,b(0,m) R (R m) n+m (f,r). Since M (f,r) M p (f,r) Ce R / R, we find that D n f,b(0,m) C er R / (R m) n+ for all n N. Letting R = n we get D n f,b(0,m) C e n n n+ ( m n )n+ for all n > m. Now Stirling s formula πne n n n (n ) and the fact ( m n )n+ e m (n ) imply that the sequence { D n f,b(0,m) } n is bounded, so that f cannot be irregular. The following auxiliary result, which can be found in [3, Lemma.], will be needed in the sequel. Lemma 8. Let 0 < α and β R. Then there is a constant C > 0 such that n=0 r αn α β Cr (n+) β e αr for all r > 0. () α Let p, by the Hausdorff Young inequality (see, for example, []) we obtain that ( z n ) ( ) M p an,r an q rqn /q, () q where q := p is the conjugate exponent of p. p Our second result provides the construction of a D-distributionally irregular entire function having, in some sense, prescribed control on their Taylor coefficients. Theorem 9. Assume that {ω n } n [0,+ ) is a sequence with lim ω n = +. Then there exists an entire function f satisfying the following properties: 4

5 (a) f n (0) ω n for all n, and (b) f is D-distributionally irregular in H(C). Proof. Firstly, the radius of convergence of the series n= n+n zn is R = lim n + n (n+) +n+ / /(n+)! = +. Therefore it converges at every z C. In particular, we have lim n +n k n=k R n = 0 for any R > 0. () Let us introduce some notation. Define ω n := min{ω n,n} (n ). Since ω n for all n N, we have obtain that, for any selection of the set S N, (n )! the expression n S ω n zn defines an entire function. The core of the proof is the search for an entire function f having large derivatives D j f for many indexes j and, simultaneously, having small derivatives D j f for other (many) indexes j. Thanks to (), we can select α N such that n=α n + n=α n + n <. Now, we choose β N with β > α. From (), again, we obtain an α N with α > β such that n n <. Choose β N with β > α. Proceeding in this way, we can recursively obtain two sequences {α} n and {β} n of natural numbers satisfying α < α < β < β < < α n < α n < β n < β n < α n+ < and Now, define the sets n + n n=α N N n < N for all N N. () A := {α n,α n +,...,αn } and B := {β n,β n +,...,βn }. n= n= 5

6 Notice that dens(a) = limsup limsup = limsup card(a {,...,n}) n card(a {,...,αn }) α n card({α n,α n +,...,α n }) α n αn α n + = lim =. αn Hence dens(a) = and, analogously, dens(b) =. Next, we set, for n N: and define the entire function ω n := { ωn if n B 0 otherwise f(z) = n= ω n z n = z n ω n. n B Clearly, f (n) (0) = ω n ω n ω n for all n. Therefore, our only task is to prove that f is distributionally irregular for D in H(C). With this aim, fix any m N. If n B then D n f,b(0,m) f (n) (0) = ω n = min{ω n,n}, which entails that lim Dn f n B,B(0,m) =. Finally, in order to show that lim Dn f = 0 it is enough to prove that n A lim f(n) n A,B(0,m) = 0 for every m N. So, fix m N as well as an ε > 0. Choose N 0 N with N 0 m and /N 0 < ε. Denote j 0 := α N0. For every j A with j j 0 there is (a unique) N N 0 such that α N j αn. Then we obtain f (j) (z) = ωnn(n )(n ) (n j +) zn = n=j n=α N ωn zn n(n )(n ) (n j +), because ωn = 0 if n B. From (), the triangle inequality and the fact αn > α N we get for every z B(0,m) that f (j) (z) ωnn(n )(n ) (n j +) mn n=α N n=α N n + n n=α N ωn mn nj N n n=α N n n n < N N 0 < ε. 6 N n

7 Therefore f (j),b(0,m) < ε whenever j A and j j 0. In other words, f (j),b(0,m) = 0, and we are done. limj j A Remarks 0.. The idea of making large gaps in the sequence of Taylor coefficients, given in the previous proof, is inspired by the construction of scrambled sets for weighted backward shifts on Köthe sequences spaces due to Wu [7] and Wu et al. [8].. Observe that the function f constructed in the proof of Theorem 9 is D- distributionally irregular in a sense stronger than the one given in Definition 6, because the set B satisfying lim n B Dn f,b(0,m) = holds for any m N. 3. Observe that the sequences {α n } n and {β n } n (hence the sets A and B) are independent of the sequence {ω n } n. This fact will be exploited in the next section. Now, we are ready to present the main result of this section. Theorem. Let p, and set a =. Then, for every ε > max{,p} 0, there exists a distributionally irregular entire function f for the differentiation operator acting on H(C) such that for some constant C > 0. M p (f,r) C er r a ε (r > 0) Proof. Since we need only prove the result for p. M p (f,r) M (f,r) for p <, Then fix p as well as an ε > 0 and denote, as usual, by q the conjugate exponent of p. Take ω n := n ε (n ). Of course, we have ω n +. By Theorem 9 there exists a D-distributionally irregular entire function f satisfying (f(0) = 0 and) f (n) (0) n ε for all n. Finally, making use of the Hausdorff-Young inequality and Lemma 8 (with α = q and β = εq), we have that M p (f,r) ( n= f (n) (0) ( rn )q ) q ( n= n εq ( rn )q ) q C er for all r > 0 and some positive constant C, which proves the theorem. r p r a ε ε = C er The following figure shows our present knowledge of possible or impossible rates e r for distributionally irregular entire function for differentiation operator. r a 7

8 a No 4 Yes? p a = p Problem. For each p [, ], give the critical order of growth between er r and er (with a = ) of a distributionally irregular entire function for the r a ε max{,p} differentiation operator. 3 Dense linear manifolds of distributionally irregular functions The study of lineability, that is, the search of linear (or, in general, algebraic) structures within nonlinear sets has become a trend in the last two decades, see e.g. the survey []. In particular, if X is a topological vector space (over R or C) and S is a subset of X, then S is called dense-lineable in X provided that there exists a dense vector subspace M of X such that M S {0}. If, in addition, M can be found with dim(m) = dim(x), then S is said to be maximal dense-lineable in X. In this section, we consider the lineability of the family of D-distributionally irregular entire functions having growth restrictions. To this end, we will need the next lemma, whose content can be found in [] (see also [, 7, 8, 0]). Following [], if A, B are subsets of a vector space, then we say that A is stronger than B whenever A+B A. Lemma 3. Assume that X is a metrizable topological vector space. Let A X. Suppose that there exists a dense vector subspace B X such that A is stronger than B and A B =. Suppose also that A {0} contains a vector subspace whose dimension equals dim(x). Then A is maximal dense-lineable. Denote by C the class of functions satisfying simultaneously all properties and growth restrictions considered in the previous section. More precisely, we denote C := {f H(C) : f is D-distributionally irregular in H(C) and sup r>0 r max{,p} ε e r M p (f,r) < for all ε > 0 and all p [, ]}. The following theorem reveals a rich linear structure inside this(seemingly small) class. Theorem 4. The set C is maximal dense-lineable in H(C). 8

9 Proof. We apply Theorem 9 to the sequence ω n := (log(n+)) t, where t > 0. By the construction given in the proof of Theorem 9 (whose notation we keep here) and Remark 0.3, there are subsets A and B of N with maximal upper density satisfying that, for each t > 0, the entire function f t (z) := n B min{n,(log(n+)) t } zn isd-distributionallyirregularin H(C). Infact,ononehand,wehavethat f (n) t (0) = min{n,(log(n+)) t } for all n B and, on the other hand, the proof of Theorem 9 allows to obtain that lim n A Dn f t,b(0,m) = 0 for all m N and all t > 0. Observe first that the functions f t (t > 0) are linearly independent. Indeed, assume that c,...,c s are complex numbers (with s and c s 0) and that 0 < t < < t s. If the linear combination F := s c k f tk (3) k= is identically zero then, after derivation, we get F (n) (0) = s c k f (n) t (0) s = c k f (n) t (0) s = c k min{n,(log(n+)) t k } (n B). k= k= Since (log(n+)) t k (log(n+)) t s n (n N; k =,...,s) and + (log(n+)) ts as n, one derives that F (n) (0) = s k= c k log(n + )) t k for n B large enough. Hence F (n) (0) + as n (n B), which is absurd. Then F is not identically 0, which shows the linear independence of {f t : t > 0}. Define M := span{f t : t > 0}. According to the previous paragraph, M is a vector subspace of H(C) with dim(m) = c = dim(h(c)), where c denotes the cardinality of the continuum. Fix any F M \ {0}. Then F has the form (3), with the c k s and the t k s as above. Therefore, for given m N, we have lim Dn F n B,B(0,m) lim F(n) (0) = +. In addition, by the triangle n B inequality, lim D n F,B(0,m) n A s k= k= c k lim D n f tk,b(0,m) = 0. n A This shows that F is D-distributionally irregular. Now, given ε > 0, there is a constant K = K(ε,c,...,c s,t,...,t s ) (0,+ ) such that F (n) (0) Kn ε for all n. The approach of the final part of the proof of Theorem yields the existence of positive constants C = C ε,p (p [, ]) satisfying e r M p (F,r) C ε,p for all r > 0. r max{,p} ε In other words, F C. Thus, our class C contains, except for 0, a vector space having maximal dimension. 9

10 Finally, let X := H(C), A := C and B := {complex polynomials}. Recall that B is a dense vector subspace of H(C). Moreover, given P B, one has P (n) = 0 for n large enough. It follows, trivially, that P is not D-distributionally irregular but P + f is D-distributionally irregular if f is. In addition, it is an easy exercise to prove that, for every b R and every p [, ], the set {f H(C) : sup r>0 r b e r M p (f,r) < } is a vector space containing the polynomials. Consequently, A B = and A+B A. An application of Lemma 3 yields the maximal dense-lineability of C. 4 Weighted Banach spaces of entire functions In this brief section, we establish the existence of distributionally irregular vectors for the differentiation operator acting on certain weighted Banach spaces of entire functions. As a sub-product, large linear manifolds consisting of such vectors will be obtained again. A weight v on C will be is a strictly positive continuous function on C which is radial, i.e. v(z) = v( z ) (z C), such that v(r) is non-increasing on [0, ) and satisfies lim r r m v(r) = 0 for each m N. We define, for p and a weight function v, the following spaces as in []: B p, = B p, (C,v) := {f H(C) : supv(r)m p (f,r) < } r>0 and B p,0 = B p,0 (C,v) := {f H(C) : lim v(r)m p (f,r) = 0}. r These spaces are Banach spaces with the norm f p,v = f p,,v := supv(r)m p (f,r). r>0 According to [, Theorem.], the polynomials are contained in B p,0 for all p and form a dense subset in it. In particular, each space B p,0 is separable. It is worth noting that, if X is a Banach space, then an operator T B(X) happens to be distributionally chaotic in the sense of Schweizer and J. Smítal [5] if and only if T has a distributionally irregular vector [, Theorem ]. Theorem 5. Let v be a weight function such that lim r v(r) er rp = 0 for some p. If the differentiation operator D : B p,0 B p,0 is continuous, then there is a dense vector subspace of B p,0 all of whose nonzero functions are distributionally irregular for this operator. Proof. By [5, Theorem.3], if v is a weight function such that lim r v(r) er for some p and D : B p,0 B p,0 is continuous, then D is frequently hypercyclic. Moreover, the set X 0 of the polynomials is a dense subset in B p,0 such that D n f tends to 0 in B p,0 for all f X 0. On the other hand, Bayart and Ruzsa have recently proved [4, Corollary 5] that if X is a Banach space and T B(X) is a frequently hypercyclic operator such 0 rp = 0

11 that T n tends pointwise to 0 on a dense subset of X then T is distributionally chaotic. Hence D is distributionally chaotic on B p,0. And since, once again, there exists a dense set X 0 such that D n f tends pointwise to 0 for all f X 0, then by [, Theorem 5] D admits a dense vector subspace consisting (except for zero) of distributionally irregular functions in B p,0. Remark 6. For any b R, let be the weight v(r) := r b e r for r b, which v(r) is non-increasing and satisfies that sup r>0 <. Then, according to [5, v(r+) Proposition.], the operator D : B p,0 B p,0 is continuous. As a consequence of Theorem 5 we obtain that for every p [,) and every ε > 0 there exists an entire function f such that M p (f,r) C er r p ε (r > 0) for some constant C > 0 as well as two increasing sequences of integers A = (n k ) k and B = (m k ) k such that dens(a) = dens(b) =, satisfying that lim k D n k f = 0 in B p,0 (hence D n k f 0 in the topology of H(C), because convergence in B p,0 implies uniform convergence on compacta) and lim k D m k f p,v =. But we do not know whether there exists m N such that lim k D m k f,b(0,m) = ; that is, we do not know whether such an f is D-irregular in H(C), so we have not obtained an improvement of Theorem. Acknowledgements. The first author is partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-7 Grant P08-FQM and by MEC Grant MTM C0-0. The second author is partially supported by MEC and FEDER, project no. MTM P. References [] R. Aron, F.J. García-Pacheco, D. Pérez-García and J.B. Seoane-Sepúlveda, On denselineability of sets of functions on R, Topology 48 (009) [] F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (006), [3] F. Bayart and E. Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics 79, Cambridge University Press, Cambridge, 009. [4] F. Bayart and I.Z. Ruzsa, Difference sets and frequently hypercyclic weighted shifts, Ergodic Theory Dynam. Systems, to appear, DOI: [5] B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North- Holland, Amsterdam, 988. [6] T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl. 373 (0), [7] L. Bernal-González, Dense-lineability in spaces of continuous functions, Proc. Amer. Math. Soc. 36 (008)

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13 [6] S.A. Shkarin, On the growth of D-universal functions, Moscow Univ. Math. Bull. 48 (993), no. 6, [7] X. Wu, Maximal distributional chaos of weighted shift operators on Köthe sequence spaces, Czechoslovak Math. J. 64 (39) (04), [8] X. Wu, G. Chen and P. Zhu, Weighted backward shift has invariant distributionally scrambled subsets, Proc. Amer. Math. Soc., to appear. Luis Bernal-González Departamento de Análisis Matemático Universidad de Sevilla Facultad de Matemáticas, Apdo. 60 Avda. Reina Mercedes, 4080 Sevilla, Spain Antonio Bonilla Departamento de Análisis Matemático Universidad de La Laguna C/Astrofísico Francisco Sánchez, s/n 387 La Laguna, Tenerife, Spain 3