Holomorphic functions having large images under the action of differential operators

TITLE: HOLOMORPHIC FUNCTIONS HAVING LARGE IMAGES UNDER THE ACTION OF DIFFERENTIAL OPERATORS. AUTHOR: LUIS BERNAL-GONZ´ ALEZ. AFFILIATION: DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. FOOTNOTES TO THE TITLE: *This work is supported in part by D.G.E.S. grant PB96-1348. 1991 Mathematics Subject Classification: Primary 30E10. Secondary 46E10, 47E05. Key words and phrases: Entire function of subexponential type, infinite order differential operator, residual set, holomorphic function, relatively compact sequence, linear metric space, large images. 1 ABBREVIATED TITLE: OPERATORS AND LARGE IMAGES. NAME AND MAILING ADDRESS OF THE AUTHOR TO WHOM PROOFS SHOULD BE SENT: LUIS BERNAL-GONZ´ ALEZ. DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. 2 HOLOMORPHIC FUNCTIONS HAVING LARGE IMAGES UNDER THE ACTION OF DIFFERENTIAL OPERATORS By LUIS BERNAL–GONZ´ ALEZ* Abstract. We prove in this note that, given a simply connected domain Gin the complex plane and a sequence of infinite order linear differential operators generated by entire functions of subexponential type satisfying suitable conditions, then there are holomorphic functions fon Gsuch that the image of any open subset under the action of those operators on fis arbitrarily large. This generalizes an earlier result about images of derivatives. A known statement about close orbits is also strengthened. 1. INTRODUCTION AND NOTATION In this paper we denote, as usual, by Nthe set of positive integers, by Cthe field of complex numbers and by D(a, r) the open disk {z∈C:|z−a|< r}. If Gis a nonempty open subset of C, then H(G) will stand for the Fr´echet space of holomorphic functions on G, endowed with the topology of the uniform convergence on compact subsets. A domain (=nonempty open subset) G⊂Cis said to be simply connected whenever its complement with respect to the extended complex plane is connected. Recall that, by Runge’s theorem [4, pp. 92-97], if Gis a simply connected 3 domain then the set of polynomials is dense in H(G). If A⊂Cis a subset with at least one finite accumulation point and Gis simply connected, then the linear manifold HA= span {ea:a∈A} is dense in H(G) (see, for instance, [7, p. 97], [6, pp. 259-260] and [2, Section 4]). We have denoted here ea(z) = exp(az) (z∈C). The diameter of a subset A⊂C is diam (A) = sup{|z−w|:z, w ∈A}. Every entire function Φ(z) = ∑∞ j=0 cjzjgenerates a formal “infinite order linear differential operator” with constant coefficients given by Φ(D) = ∑∞ j=0 cjDj, where Ddenotes the differentiation operator Df =f′and D0=I= the identity operator. The function Φ is said to be of exponential type if and only if there are constants K1, K2∈(0,+∞) such that |Φ(z)| ≤ K1eK2|z|(z∈C). Φ is said to be of subexponential type if and only if for every ε > 0 there is a constant K=K(ε)∈(0,+∞) such that |Φ(z)| ≤ Keε|z|(z∈C). Each function of subexponential type is obviously of exponential type. With essentially the same methods of Valiron [9, p. 35] (see also [3, pp. 58-60] and [2, Theorem 5]), it can be proved that Φ(D) is a well-defined operator on H(G) as soon as Φ 4 is of subexponential type (and, in fact, on H(C) if Φ is just of exponential type). The reader is referred to [8] for a systematic study of this kind of operators. From now on, if Xis a linear metric space, then we denote ||x|| =d(x, 0) for x∈X, where dis the (translation-invariant) metric of X. For instance, the translation-invariant metric d(f, g) = ∞ ∑ n=1 1 2n·supKn|f−g| 1 + supKn|f−g|(f, g ∈H(G)) generates the natural topology of G. Here (Kn) is an exhaustive nondecreasing sequence of compact subsets of G. In 1987, R. M. Gethner and J. H. Shapiro [5], when studying the existence of universal vectors for certain kinds of operators on spaces of holomorphic functions, proved the following related result [5, Theorem 2.4]. THEOREM A. Suppose Tis a continuous linear operator on a separable complete linear metric space Xand Tnx→0 (n→ ∞)for every x∈ D, where D is a dense subset of Xand Tn=T◦T◦ · · · ◦ T(ntimes). Let (xn)be a sequence in Xsuch that xn→0 (n→ ∞). Then the set of vectors x∈Xfor which lim infn→∞ ||Tnx−Tnxn|| = 0 is a dense Gδsubset of X. As an application of Theorem A to function theory, it is shown in [5, Theorem 3.7] the next theorem, that establishes the existence of entire functions for which many derivatives have “large images” on prefixed arbitrarily small open subsets. 5 THEOREM B. Suppose (ρn)is an unbounded, increasing sequence of positive numbers for which limn→∞ ρ1/n n n= 0. Then there exists a dense Gδsubset Mof H(G)satisfying that, for every f∈Mand every nonempty open set V⊂G, there are infinitely many n∈Nsuch that f(n)(V)⊃D(0, ρn). Our aim in this note is to extend the latter result to certain kinds of infinite order differential operators on simply connected domains. In passing, Theorem A can also be manifestly strengthened. 2. SETS OF POINTS CLOSE TO THE ORBIT OF A RELATIVELY COMPACT SEQUENCE We start with an elementary lemma. Let Xbe a topological space, (Y, d) a metric space and denote, as usual, by C(X, Y ) the space of all continuous mappings from Xinto Y. If σ= (sn) and τ= (Tn) are sequences in Xand C(X, Y ) respectively, then we put M(σ, τ) = {x∈X: lim inf n→∞ d(Tnx, Tnsn) = 0}. LEMMA 1. If X,(Y, d),σ= (sn)and τ= (Tn)are as before, then M(σ, τ)is aGδsubset of X. Proof. Fix a point x∈X. Observe that x∈M(σ, τ) if and only if for each 6 pair N, k ∈Nthere is n > N such that d(Tnx, Tnsn)<1/k, that is, M(σ, τ) = ∩ N∈N∩ k∈N∪ n>N T−1 n(Gn,k), where Gn,k ={y∈Y:d(y, Tnsn)<1/k}, which is an open ball in X. Since Tnis continuous, T−1 n(Gn,k) is open in Xand so M(σ, τ) is a Gδsubset. //// For every sequence σ= (sn) in X, denote LP (σ) = {α∈X:αis a limit point for (sn)}. LP (σ) may well be empty. If Xand Yare topological vector spaces, then L(X, Y ) will stand for the subspace of C(X, Y ) of all linear mappings from Xinto Y. Recall that a subset in a Baire space is residual if and only if it contains a dense Gδsubset. The next result generalizes Theorem A. THEOREM 1. Assume that Xand Yare linear metric spaces, in such a way that Xis a Baire space. Let σ= (sn)and τ= (Tn)be sequences in Xand L(X, Y ), respectively. Suppose that the following three conditions are satisfied: (a) σis relatively compact. (b) limn→∞ Tnα= 0 for every α∈LP (σ). (c) There exists a dense subset D ⊂ Xsuch that lim infn→∞ ||Tnx|| = 0 for all x∈ D. Then M(σ, τ)is residual in X. 7 Proof. Note that, by (a), LP (σ) is not empty. Recall that ||x|| =d1(x, 0) for every x∈Xand ||y|| =d(y, 0) for every y∈Y, where d1, d are the metrics on X, Y (resp.), which are translation-invariant. By Lemma 1, M(σ, τ) is a Gδsubset of X. Let us keep in mind the notation of the proof of that lemma. Since Xis Baire and the sets S(N, k) := ∪ n>N T−1 n(Gn,k) (N, k ∈N) are open, it suffices to show that every S(N, k) is dense in X. For this, fix N, k ∈N, a point x0∈ D and ε > 0. By (c), there is a sequence n1< n2< ... < nj< ... of positive integers such that lim j→∞ ||Tnjx0|| = 0. But σis relatively compact, so there is a point α∈Xand a subsequence m1< m2< ... < mj< ... of (nj) such that lim j→∞ ||smj−α|| = 0. From (b), we have that lim j→∞ ||Tmjα|| = 0. In particular, there exists n>N such that ||Tnx0|| <1 2k,||Tnα|| <1 2kand ||sn− α|| < ε. Define the point x=x0+sn−α. Then ||x−x0|| =||sn−α|| < ε and, by linearity, d(Tnx, Tnsn) = ||Tnx−Tnsn|| = ||Tnx0−Tnα+Tnsn−Tnsn|| ≤ ||Tnx0|| +||Tnα|| <1 2k+1 2k=1 k. 8 Thus, x∈S(N, k)∩ {z∈X:d1(z, x0)< ε}, that is, every point x0∈ D is in the closure of S(N, k). But Dis dense in X. Consequently, S(N, k) is also dense in X, as required. //// 3. DIFFERENTIAL OPERATORS AND LARGE IMAGES In this section we extend Theorem B to differential operators. Recall that if Φ(z) = ∑∞ j=0 cjzjis a nonconstant entire function, then its multiplicity for the zero at the origin is m= min{j∈ {0,1,2, ...}:cj= 0}. We start with the following easy lemma, whose proof is omited since it is a simple calculation. LEMMA 2. If a, b are complex numbers with a= 0, and m∈N, then (aDm+bDm+1)(1 a zm+1 (m+ 1)! −b a2 zm m!) = z(z∈C). THEOREM 2. Let Gbe a simply connected domain of C,Φn(z) = ∑∞ j=0 c(n) jzj (n∈N)nonconstant entire functions of subexponential type and (ρn)an unbounded sequence of positive numbers. Denote by m(n)the multiplicity of Φnfor the zero at the origin. Suppose that the following conditions are fulfilled: (1) The sequence (m(n)) is unbounded. (2) max{lim supn→∞ 1 1+m(n)(ρn |c(n) m(n)|)1 1+m(n),lim supn→∞ 1 1+m(n)(ρn|c(n) m(n)+1| |c(n) m(n)|2)1 1+m(n)} ≤1 e·diam (G). Then there exists a residual subset Mof H(G)satisfying that, for every f∈M 9 acumulation point and B(1 + |c0|)\ {0}is not empty. Observe again that cmis the only coefficient relevant to the conclusion of the next result. Theorem 4 also contains Theorem B as a special case. THEOREM 4. Let Gbe a simply connected domain of the complex plane, Φ(z) = ∑∞ j=0 cjzja nonconstant entire function of subexponential type with multiplicity mand (ρn)an unbounded sequence of positive numbers. Suppose that one of the following properties is satisfied: (1) c0= 0 and lim supn→∞ ρn |c0|n= 0. (2) c0= 0 and lim supn→∞ ρ1/n n nm≤(m e·diam (G))m· |cm|. Then the same conclusion of Theorem 2 holds. Only a remark before the end. Fix N∈N. By considering sequences of functions of the form fn(z) = α(n, ε)(z−w)Nor of the form fn(z) = α(n, ε)[ec(z−w)− ∑N−1 j=0 cj(z−w)j j!] (n∈N), where cand α(n, ε) are appropriate constants, the interested reader (if any) could try to show that, under suitable conditions on the Taylor coefficients c(n) jof the entire functions Φn(or on the coefficients cjof a single Φ), there exists a residual subset M⊂H(G) with the following property: for each member f∈Mand each nonempty open subset V⊂G, there are infinitely many n∈Nfor which the equation f(n)(z) = whas at least Nsolutions in Vfor every w∈D(0, ρn). 16 REFERENCES 1. L. V. Ahlfors, “Complex Analysis”, McGraw-Hill, London, 1979. 2. L. Bernal-Gonz´alez, Hypercyclic sequences of differential and antifferential operators, J. Approx. Theory, to appear. 3. D.G. Dickson, “Expressions in series of solutions of linear difference-differential and infinite order differential equations with constant coefficients”, Memoirs of the Amer. Math. Soc. 23, Providence, Rhode Island, 1957. 4. Gaier, “Lectures on complex approximation”, Birkhauser, Boston, 1987. 5. R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288. 6. G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. 7. L. Hormander, “An Introduction to Complex Analysis in Several Variables”, Van Nostrand, Princeton, NJ, 1966. 8. A. Martineau, ´ Equations diff´erentielles d’ordre infinie, Bull. Soc. Math. France 95 (1967), 109-154. 9. G. Valiron, Sur les solutions des ´equations differentielles lin´eaires d’ordre infinie et `a coefficients constants, Ann. ´ Ecole Norm. (3) 46, 25-53 (1929). Luis Bernal-Gonz´alez Departamento de An´alisis Matem´atico Facultad de Matem´aticas Avenida Reina Mercedes. Apartado 1160 41080 Sevilla (S p a i n) E-mail: lb[email protected] 17