Holomorphic T-monsters and strongly omnipresent operators

TITLE: HOLOMORPHIC T–MONSTERS AND STRONGLY OMNIPRESENT OPERATORS. 1 FIRST 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]. SECOND AUTHOR: MAR´ IA DEL CARMEN CALDER ´ ON–MORENO. AFFILIATION: DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E–MAIL: [email protected]. 1FOOTNOTES TO THE TITLE: This work has been partially supported by DGES grant PB96–1348 and the Junta de Andaluc´ıa. 1991 Mathematics Subject Classification: Primary 30E10. Secondary 47B38, 47E05. 1 ABBREVIATED TITLE: HOLOMORPHIC T–MONSTERS. NAME AND MAILING ADDRES 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 T-monsters and strongly omnipresent operators by L. BERNAL–GONZ´ ALEZ and M.C. CALDER´ ON–MORENO April 22, 2019 Abstract Assume that Gis a nonempty open subset of the complex plane and that Tis an operator on the linear space of holomorphic functions in G, endowed with the compact–open topology. In this paper we introduce the notions of strongly omnipresent operator and of T–monster, which are related to the wild behaviour of certain holomorphic functions near the boundary of G.T–monsters extend a concept introduced by W. Luh and K.–G. Grosse–Erdmann. After showing that Tis strongly omnipresent if and only if the set of T–monsters is residual, it is proved in this paper that certain kinds of infinite order differential and antidifferential operators are strongly omnipresent, which improves some earlier nice results due to the mentioned authors. Key words and phrases: holomorphic monster, T–monster, strongly omnipresent operator, infinite order differential operator, infinite order antidifferential operator, entire function of subexponential type, affine linear mappings, Laplace transform. 1 Introduction and notation Throughout this paper Gwill stand for a nonempty open set in the complex plane C.Nis the set of positive integers, N0= N∪ {0},Zis the set of all integers, C∞is the extended complex 3 plane C∪ {∞},∂G is the (finite) boundary of G,B(a, r) (B(a, r)) is the euclidean open (closed, respectively) disk with center aand radius r(a∈C, r > 0) and D=B(0,1). If A⊂Cthen A0is the interior of Aand we denote LT(A) = {affine linear transformations τ(z) = az +bsuch that τ(D)⊂A}. H(G) denotes, as usual, the linear space of holomorphic functions on G, endowed with the topology τuc of uniform convergence on each compact subset in G. Let K(G) be the family of compact subsets of G. It is known that the family {D(f, K, ) : f∈H(G), K ∈ K(G),  > 0} where D(f, K, ) = {g∈H(G) : |g(z)−f(z)|<  for all z∈K} is a basis for τuc. If Kis a compact set of C, then A(K) will stand for the linear space C(K)∩H(K0), which becomes a Banach space if it is endowed with the maximum norm. A topological space Xis a Baire space if and only if the intersection of a countable family of open dense subsets is also dense. Baire’s theorem asserts that each completely metrizable topological space is a Baire space. Consequently, H(G) is a Baire space. In a Baire space X, a subset is residual when it contains a dense Gδ–subset of X; such a subset is “very large” in X. These notions and results can be found, for instance, in [13, pp. 213–214], and [21, pp. 40–41]. If f∈H(G) and j∈N0we denote, as usual, by f(j)the derivative of fof order j. The linear operator Dj:H(G)→H(G) defined by Djf=f(j)is continuous. 4 An entire function Φ(z) = Pj≥0ajzjon Cis said to be of exponential type whenever there exist positive constants Aand Bsuch that |Φ(z)| ≤ AeB|z|for all z∈C. This happens if and only if lim sup j→∞ (j!|aj|)1 jis finite (cf. [23, Chapter VII]). Φ is said to be of subexponential type if and only if given  > 0, there is a positive constant A=A() such that |Φ(z)| ≤ Ae|z|for all z∈C. It happens that Φ is of subexponential type if and only if lim sup j→∞ (j!|aj|)1 j= 0 (cf., e.g., [5, 2.2.9–11]). Trivially, each entire function of subexponential type is also of exponential type. To every entire function Φ we can associate a “formal” infinite order differential operator with constant coefficients L= Φ(D), that is, L=P∞ j=0 ajDjwith D0=I= the identity operator. The following statement is easy to prove (see [3], [7, pp. 58–60], [11, Section 5] and [25, p. 35]) and furnishes a sufficient condition in order that Φ(D) can be an operator. Theorem 1.1 Let Φ(z) = P∞ j=0 ajzjan entire function of subexponential type. Then ∞ X j=0 |aj|sup z∈L |f(j)(z)|<+∞for every L∈ K(G) and every f∈H(G), and Φ(D)is a well–defined continuous line- ar operator on H(G). If G=C, the same assertion holds just by assuming that Φis of exponential type. Recall now the “infinite order antidifferential operators”. They were studied by the first author (see [4]). Firstly, assume that G⊂C is a simply connected domain and that ais a fixed point in G. If 5 j∈Nand f∈H(G), denote by D−jfthe unique antiderivative gof order jof f(i.e., (D−jf)(j)=f) such that g(k)(a) = 0 (k= 0,1, . . . , j −1). In fact, we have D−jf(z) = Zz af(t)(z−t)j−1 (j−1)! dt (z∈G) where the integral is taken along any rectifiable curve γ⊂Gjoining ato z. Each D−jis a continuous linear operator on H(G). If δ∈[0,+∞), then we denote by S(δ) the set of formal complex power series Ψ(z) = P∞ j=0 cjzjsuch that lim sup j→∞ |cj| j!! 1 j ≤δ. For fixed c∈Gwe use the notation ∆(c, G) = sup z∈G inf{r > 0 : cis in the connected component of B(z, r)∩Gcontaining z}(see [2, 6]). The following result can be found in [4, Theorem 6]. Theorem 1.2 If Ψ(z) = ∞ X j=0 cjzj∈S 1 ∆(a, G)!then the series Ψ(D−1) = ∞ X j=0 cjD−jdefines a continuous linear operator on H(G). W. Luh has studied in a series of papers the problem of existence of functions with wild behaviour at every boundary point of certain open sets (see [14, 15, 16 and 17]). The following strong result can be found in [17]. Theorem 1.3 Let G⊂C,G6=C, be an open set with simply connected components. Then there exists a function f∈H(G)with the following properties: (1) For every t∈∂G, every compact subset Kwith connected complement and every g∈A(K), there exist affine linear mappings τn(z) = anz+bnwith τn(K)⊂G(n∈N)and an→0, 6 bn→t(n→ ∞)such that f(τn(z)) →g(z)(n→ ∞) uniformly on K. (2) In addition, each derivative f(j)(j∈N)of fand each antiderivative of fof arbitrary order has the boundary behaviour described in (1). Functions satisfying (1) and (2) are called “holomorphic monsters” by Luh. He shows in [17] that its set is dense in H(G). In addition, he proves that every fin H(G) satisfying (1) also satisfies the next two properties: (a) For every bounded open set U⊂Cwith simply connected components, every g∈H(U) and every t∈∂G, there exist affine linear mappings τn(z) = anz+bnwith τn(U)⊂G(n∈N) and an→0, bn→t(n→ ∞) such that f(τn(z)) →g(z) in H(U). (b) For every bounded Lebesgue–measurable set S⊂C, every Lebesgue–measurable function g:S→C∞and every t∈ ∂G, there exist affine linear mappings τn(z) = anz+bnwith τn(S)⊂G(n∈N) and an→0, bn→t(n→ ∞) such that f(τn(z)) →g(z) almost everywhere in S. The properties exhibited in Theorem 1.3 and in (a) and (b) can be expressed in terms of the maximality of certain generalized cluster sets introduced in [17]. A little later, in 1987, K.–G. Grosse–Erdmann showed [12, Kapitel 3] that if f∈H(G) then it is a monster in the sense of Luh if and only if every derivative and every antiderivative of fof arbitrary order, say F, satisfies that for each Jordan domain Ω ⊂C, each t∈∂G and each g∈H(Ω), there exist two sequences {an}n∈N 7 and {bn}n∈Nin Csuch that an→0, bn→t(n→ ∞), anz+bn∈G (z∈Ω, n∈N) and F(anz+bn)→g(z) (n→ ∞) uniformly on compact subsets in Ω. In fact [12, Satz 3.0.2], it suffices to take Ω = Din the latter property. By using the fact that being a monster is equivalent to the universality with respect to a certain family of composition–differentiation–antidifferentiation operators, he also proves [12, Satz 3.1.8] that the set of monsters on Gis not only dense but residual in H(G). Monsters with additional properties were constructed by Luh [18] and I. Schneider [24], see also [19]. On the other hand, the first author [1] introduced in 1992 the “omnipresent holomorphic operators”. Let us recall this notion. Let Tbe a continuous mapping T:H(G)→H(G). Tneed not be linear. Denote O(∂G) = {V⊂C:Vis open and V∩∂G 6=∅}. Recall that Tis said to be omnipresent if each subset R(T, V, W) = {f∈H(G) : there exists z∈G∩Vsuch that Tf(z)∈W}is dense in H(G) for all V∈O(∂G) and all nonempty open subsets W⊂C. In [1] it is shown that each differential and each antidifferential operator is omnipresent. These properties will be strengthened in Theorem 3.1 and, partially, in Theorem 4.2. In this paper we introduce two original concepts which are connected with each other: the T–monsters and the strongly omnipresent operators. With these, we will be following the ways opened by Luh and Grosse–Erdmann from the different point of view presented by the first author in [1]. In particular, operators of type Φ(D) and Ψ(D−1) are going to be studied. 8 2 Definitions From now on, we denote by Gan arbitrary open subset of C, G6=C. Let T:H(G)→H(G) be a continuous operator, which need not be linear, as it was supposed before. If g∈H(D),  > 0, r∈(0,1) and V∈O(∂G), then U(T, g, , r, V ) will stand for the set U(T, g, , r, V ) = {f∈H(G) : there exists τ∈LT(V∩G) such that |(Tf)(τ(z)) −g(z)|<  for all z∈rD}. It is evident that each of these sets is open in H(G). We will say that Tis strongly omnipresent in Gif and only if every set U(T, g, , r, V ) (g∈H(D),  > 0, r∈(0,1), V∈O(∂G)) is dense in H(G). It is evident that each strongly omnipresent operator on H(G) is also omnipresent. By Theorem 1(c) of [1] –applied on T= the identity operator and g(z) = exp z– we have that f7→ exp fis a (nonlinear) omnipresent operator on H(G) which is not strongly omnipresent by Hurwitz’s theorem and Lemma 2.1 together with Theorem 2.2 below. Up to date, we do not know whether a linear omnipresent non-strongly omnipresent operator can exist or not. Inspired by [12], we will say that a function f∈H(G) is a T– monster in Gif and only if for each Jordan domain Ω ⊂C, each g∈H(Ω) and each t∈∂G, there exist two sequences {an}n∈Nand {bn}n∈Nin Csuch that an→0, bn→t(n→ ∞); anz+bn∈Gfor all n∈Nfor all z∈Ω and (Tf)(anz+bn)→g(z) (n→ ∞) 9 In order to show that Tis strongly omnipresent, fix f∈H(G),  > 0, a compact subset K⊂G,V∈O(∂G), δ > 0, r∈(0,1) and g∈H(D). It is clear that we can find b, B, γ, τ and Lsatisfying: 1. Lis compact, {a} ∪ K⊂L⊂Gand the complement of Lis connected, 2. Bis a closed ball with B=τ(D)⊂V∩G\L,τbeing a nonconstant affine linear mapping, 3. γis a rectifiable Jordan arc in Gjoining ato band γ=γ1+γ2, where γk(k= 1,2) are rectifiable Jordan arcs too, γ1=γ∩L, bis the end point of γ2and γ∩B={b}. Assume that we have proved the following property:                      The linear operator S:Hb(C)→H(B0) given by (Sψ)(w) = λψ(w) + Rw bψ(t)ϕ(w, t)dt (w∈B0) has dense range,                      (P) where Hb(C) := {ψ∈H(C) : ψ(b) = 0}, which is a closed subspace of H(C). Each integration curve may unambiguously be chosen as a rectifiable Jordan arc joining bto wcontained in B, for instance, the segment [b, w]. Denote γ(w) = γ+[b, w]. Observe that length(γ(w)) ≤length(γ)+ 2sfor all w∈B, where sis the radius of B. Define the constant α by α= sup{|ϕ(w, t)|:w∈B, t ∈γ∪B}. We can suppose that γ2has a parametrization u∈[0,1] 7→ β(u), 16 where βis injective. If ˜ f:γ→Cis defined as ˜ f(w) =    f(w) if w∈γ1 f(β(0))(1 −u) if w=β(u)∈γ2, then ˜ fis continuous on γ. Note that g◦τ−1is defined and holomorphic on B0because g is holomorphic on D=τ−1(B0). The same is true for the function w7→ Zγϕ(w, t)˜ f(t)dt. From property (P), there exists an entire function Ψ such that Ψ(b) = 0 and (SΨ)(w)−g(τ−1(w)) −Zγϕ(w, t)˜ f(t)dt<δ 3for all w∈B1 (1) where B1:= τ(rD). Let K0=L∪γ∪B. Define the function f1:K0→Cas f1(w) =          f(w) if w∈L f(β(0))(1 −u) if w=β(u)∈γ2 Ψ(w) if w∈B. Then K0is a compact set whose complement is connected, f1is continuous on K0and holomorphic in its interior K0 0=L0∪B0. By Mergelyan’s Theorem ([10, pp. 97–109] or [22, Chap. 20]), there is a polynomial Psuch that |P(z)−f1(z)|<min(, δ, 1) 1+3|λ|+ 3α(length(γ)+2s)for all z∈K0.(2) We have, for every w∈B1, (TP)(w)−g(τ−1(w)) = λP(w) + Zγ(w)ϕ(w, t)P(t)dt −g(τ−1(w)) = λ(P(w)−Ψ(w)) + Zγ(w)ϕ(w, t)(P(t)−f1(t))dt +Zγϕ(w, t)˜ f(t)dt+ 17 λΨ(w) + Zw bϕ(w, t)Ψ(t)dt −g(τ−1(w)) = I+J+M, where I=Zγ(w)ϕ(w, t)(P(t)−f1(t)) dt, J= (SΨ)(w)−g(τ−1(w)) −Zγϕ(w, t)˜ f(t)dt, M=λ(P(w)−Ψ(w)). Inequality (2) yields |I|<δ 3 and |M|<δ 3. From (1) we obtain |J|<δ 3, so |(TP)(w)−g(τ−1(w))|< δ because of the triangle inequality. A change of variables shows that |(TP)(τ(z)) −g(z)|< δ for all z∈rD.(3) But (2) also gives that |P(w)−f(w)|<  on K(4) because K⊂L⊂K0. Then (3) and (4) tell us that P∈D(f, K, )∩U(T, g, δ, r, V ). Hence U(T, g, δ, r, V ) is dense in H(G) and Tis strongly omnipresent, as required. Thus, we should prove property (P). At this point the shape ϕ(w, t) = h(w−t) of ϕis crucial. A “Laplace transform” method will be used in this part of the proof. From now on, the letter zmay be any complex number, not necessarily in D. 18 By applying Lemma 4.1 on G=B0and taking into account that every polynomial can be written as a finite linear combination of powers (z−b)m, it suffices to show that there is m0∈Nsuch that for each m≥m0a corresponding function Ψ ∈H(C) can be found in such a way that Ψ(b) = 0 and λΨ(z) + Zz bΨ(t)h(z−t)dt = (z−b)mfor all z∈C. By the analytic continuation principle and a simple change of variables, we should be done whenever we are able to prove the existence of m0∈Nsuch that for every m≥m0there is an entire solution Ψ(z) with Ψ(0) = 0 of the functional equation λΨ(x) + (Ψ ? h)(x)−xm= 0 (x∈(0,+∞)),(5) where Ψ ? h is the convolution product (Ψ ? h)(x) = Zx 0Ψ(t)h(x−t)dt. Let us choose m0= 2 + [the multiplicity of hfor the zero at the origin] (the multiplicity may be, of course, zero) and fix m≥m0. Since his of exponential type, its Laplace transform (Lh)(z) = Z+∞ 0h(t)e−ztdt is defined and holomorphic on a certain half plane {Re z > x0}. In fact, if ∞ X j=0 hjzjis the Taylor series of h, then ρ:= lim sup j→∞ (j!|hj|)1 j< +∞and the series ∞ X j=0 j!hj zj+1 converges on {|z|> ρ}. We have that (Lh)(z) = ∞ X j=0 j!hj zj+1 on Re z > ρ. We have used that the Laplace transform of each function xmis m! zm+1 . Consider the expression F?(z) = m!zm+1 λ+P∞ j=1(j−1)!hj−1zj. 19 Note that the series in the denominator defines a holomorphic function on B(0,1 ρ) whose zero at the origin has multiplicity not greater than m0−1, so F?(0) = 0 (with multiplicity at least 2) and there is µ∈(0,1 ρ) such that F?∈H(B(0, µ)). Then the function F(z) := F?(1 z) is holomorphic in the neighbourhood |z|>1 µof ∞and has a zero at this point. Consequently, it has an expansion F(z) = ∞ X j=0 cj zj(|z|>1 µ), with c0=c1= 0. Thus, Fis representable by means of an absolutely convergent Laplace integral. In fact (see [8, p. 66]), the function Ψ(z) = ∞ X j=1 cj+1 j!zj(z∈C) is entire and of exponential type, and satisfies Ψ(0) = 0 and F(z) = Z∞ 0Ψ(t)e−ztdt (Re z > R) for certain R∈(0,+∞). Hence  λ+ ∞ X j=0 j!hj zj+1  Z∞ 0Ψ(t)e−ztdt =m! zm+1 (Re z > R), that is, λ(LΨ)(z)+(Lh)(z)·(LΨ)(z)=(Lσ)(z) (Re z > R), where σ(x) := xm. But, by Borel’s theorem, L(Ψ?h)(z) = L(Ψ)(z)· (Lh)(z), so, by linearity, L(λΨ+Ψ? h −σ)(z) = 0 (Re z > R). Since λΨ+Ψ?h−σis continuous on the interval (0,+∞), we obtain λΨ + Ψ ? h −σ= 0 on that interval (see [8, pp. 53–54]). But this is (5). This completes the proof. 20 We obtain immediately the following corollary, which of course can be applied on a nonzero entire function Ψ. Corollary 4.3 Let G⊂Cbe a simply connected domain with G6= C. Fix a point a∈Gand let Ψ(z) := ∞ X j=0 cjzj∈H(B(0, R)) for some R > 0,Ψbeing not identically zero. Consider the operator T= Ψ(D−1)on H(G). Then Tis strongly omnipresent. Proof. Recall that, in the definition of Ψ(D−1), a point a∈G has been fixed. We will see that the hypotheses of Theorem 3.1 or Theorem 4.2 are satisfied. We have that lim sup j→∞ (|cj|1 j)<+∞(6) and, trivially, lim sup j→∞ |cj| j!! 1 j = 0.(7) By (7) and Theorem 1.2, Tis a linear continuous operator on H(G) and for any f∈H(G) Tf(z) = λf(z) + Zz ah(z−t)f(t)dt (for all z∈G) where λ≡c0∈Cand h(w) = ∞ X j=1 cj (j−1)!wj−1. Observe that his an entire function of exponential type, because of (6). Consequently, either his identically zero (so λ6= 0, because Ψ is not identically zero) and we get all hypotheses of Theorem 3.1 just by taking Φ(z)≡λ, or his not identically zero and we apply Theorem 4.2. In any case Tis strongly omnipresent and the proof is finished. The following final result is a straightforward consequence of Theorems 2.2, 3.1 and Corollary 4.3, and of the fact that, in a Baire 21 space, the intersection of a countable family of residual subsets is also residual. In fact, the Baire category approach is now a classical tool in the study of universality. Note that, for simply connected domains, Luh’s and Grosse–Erdmann’s results follow from the special case Φj(z) = z|j|(j∈Z). Theorem 4.4 Assume that Gis a simply connected domain of C, G6=C. Fix a point a∈Gand let D−1be the corresponding antidifferential operator. Suppose that {Φj}j∈Zis a family of nonidentically zero entire functions such that Φjis of subexponential type for j∈N0. Denote Tj= Φj(D)whenever j∈N0and Tj= Φj(D−1) whenever −j∈N. Then there is a residual subset in H(G)consisting of functions fsatisfying the following property: For each j∈Z, each t∈∂G, each compact subset K⊂Cwith connected complement and each g∈A(K), there exist affine linear transformations τnwith τn(K)⊂G(n∈N)and an→0,bn→t(n→ ∞)such that (Tjf)◦τn→g(n→ ∞)uniformly on K. ACKNOWLEDGEMENT The authors are grateful to the referees for helpful comments and suggestions. References [1] L. 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(3) 46 (1929), 25–53. LUIS BERNAL GONZ ´ ALEZ MAR´ IA DEL CARMEN CALDER ´ ON MORENO DEPARTAMENTO DE AN ´ ALISIS MATEM ´ ATICO DEPARTAMENTO DE AN ´ ALISIS MATEM ´ ATICO FACULTAD DE MATEM ´ ATICAS, APDO. 1160 FACULTAD DE MATEM ´ ATICAS, APDO. 1160 AVENIDA REINA MERCEDES AVENIDA REINA MERCEDES 41080 SEVILLA, SPAIN 41080 SEVILLA, SPAIN E–mail: lb[email protected] E–mail: [email protected] 25