Dense linear manifolds of monsters

TITLE: DENSE LINEAR MANIFOLDS OF MONSTERS. 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: The work of the two authors has been partially supported by DGES grant PB96–1348 and the Junta de Andaluc´ıa. 2000 Mathematics Subject Classification: Primary 30E10. Secondary 30H05, 47A16, 47B38. 1 ABBREVIATED TITLE: DENSE LINEAR MANIFOLDS OF MONSTERS. 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 Dense linear manifolds of monsters L. BERNAL–GONZ ´ ALEZ and M.C. CALDER ´ ON–MORENO Abstract In this paper the new concept of totally omnipresent operators is introduced. These operators act on the space of holomorphic functions of a domain in the complex plane. The concept is more restrictive than that of strongly omnipresent operators, also introduced by the authors in an earlier work, and both of them are related to the existence of functions whose images under such operators exhibit an extremely wild behaviour near the boundary. Sufficient conditions for an operator to be totally omnipresent as well as several outstanding examples are provided. After extending a statement of the first author about the existence of large linear manifolds of hypercyclic vectors for a sequence of suitable continuous linear mappings, it is shown that there is a dense linear manifold of holomorphic monsters in the sense of Luh, so completing earlier nice results due to Luh and Grosse–Erdmann. Key words and phrases: Holomorphic monster, T–monster, strongly omnipresent operator, totally omnipresent operator, dense linear manifold, hypercyclic sequence, composition operator, infinite order linear differential operator, integral operator. 1 Introduction In 1985 Luh [22] introduced the concept of holomorphic monsters. Roughly speaking, a holomorphic monster in the sense of Luh is a holomorphic function on a simply connected domain Gof the complex plane such that it and all its derivatives and antiderivatives possess an extremely wild behaviour near the boundary, see below. Luh proved the existence of adense subset of monsters in the space H(G) of holomorphic functions on G, endowed with the compact–open topology. Note that H(G) is a Fr´echet space, hence a Baire space. Two years later, Grosse-Erdmann, by using techniques of functional analysis via certain composition–differentiation–antidifferentiation operators, showed that, in fact, there exists aresidual set of monsters in H(G) [18, Kapitel 3]. In this work we will establish, among other results, the existence of a dense linear manifold of holomorphic monsters. Consequently, the set of Luh monsters is large not only topologically but also algebraically. The reader is referred to [23, 24, 27] for further interesting results on this topic. 3 As a matter of fact, we will state our main result (Theorem 5.1) in a much more general form, by means of the introduction of the notion of totally omnipresent operators, see Section 2. This notion is strictly stronger than that of strongly omnipresent operators, which we recall shortly together with the related concept of T–monsters, both of them introduced by the authors in [6]. In the present paper we strengthen (Theorem 3.1) a recent statement of the first author [4] (see Theorem 1.1 below) about the existence of large linear manifolds of hypercyclic vectors for a sequence of continuous linear mappings. Theorem 5.1 is extracted as a consequence. Furthermore, a number of practicable sufficient conditions for an operator to be totally omnipresent are furnished in Section 4, as well as a large family of examples including differential, antidifferential, integral, composition and multiplication operators. Now, we pass to fix some notations and definitions. Throughout this paper Gwill stand for a domain in the complex plane Cand ∂G will denote its boundary taken in the extended complex plane C∞=C∪ {∞}.Nis the set of positive integers, N0=N∪ {0},Zis the set of integers, Ris the real line, and B(a, r) = {z:|z−a|< r}is the euclidean open ball with center aand radius r(a∈C, r > 0). The corresponding closed ball is B(a, r). An operator always refers to a continuous (not necessarily linear) selfmapping. We denote by O(∂G) the set of all open subsets of C∞meeting ∂G. If A⊂Cthen A(A0) represents the closure (the interior, respectively) of A,kfkA:= supz∈A|f(z)|, where fis a complex function defined in A, and LT(A) is the set of all affine linear transformations τ,τ(z) = az +b, such that τ(D)⊂A, where D:= B(0,1). As for the definition of T–monsters and of its associated notion of strongly omnipresent operators, we fix here one which is slightly stronger than that of [6], because there (as in [22]) the domain Gwas never Cin order that the finite boundary be non-empty. Nevertheless, as pointed out in [9], using chordal distances, all proofs can be adapted to the case where the boundary point under consideration is the point of infinity. Thus, as in [9], we establish the following definition. Definition 1.1. (a) A function f∈H(G) is a holomorphic monster whenever the following universality property is satisfied: For each g∈H(D) and each t∈∂G there exists a sequence (τn) of affine linear transformations with τn(z)→t(n→ ∞) uniformly on Dand τn(D)⊂G(n∈N) such that f(τn(z)) →g(z) (n→ ∞) locally uniformly in D. (b) Let T:H(G)→H(G) be an operator. Then a function f∈H(G) is a T–monster if Tf is a holomorphic monster. The set of T–monsters is denoted by M(T). 4 (c) An operator T:H(G)→H(G) is strongly omnipresent if for all g∈H(D), ε > 0, r∈(0,1) and V∈O(∂G) the set U(T, g, ε, r, V ) := {f∈H(G) : there exists some τ∈LT(V∩G) such that k(Tf)◦τ−gkrD< ε} is dense in H(G). As in [6, Theorem 2.2], it is easy to prove that Tis strongly omnipresent if and only if the set M(T) is residual, i.e., its complement in H(G) is of first category (see also [1] for the weaker concept of omnipresent operators and [9, Example 3.4] for a linear example of an omnipresent operator which is not strongly omnipresent). Observe that an easy continuity argument allows us to restrict ourselves to non-constant affine linear transformations in parts (a) and (c) of the last definition. Note also that due to the results of [18, Kapitel 3] a function f∈H(G) –where Gis simply connected– is a holomorphic monster in the sense of Luh [22] (for future references, we call such an faLuh–monster) if and only if fis simultaneously aDj–monster and a D−j a–monster for all j∈N0. Here Dis the differentiation operator Df =f0,D0=Iis the identity operator, Dj+1 =D◦Dj,ais a fixed point in the simply connected domain G,D0 a=Iand, for each j∈N,D−j adenotes the unique antiderivative Fof fof order jsuch that F(k)(a) = 0 (k∈ {0,1, . . . , j −1}). Since the intersection of countably many residual sets is again residual, the existence of Luh-monsters is thus a direct consequence of the strong omnipresence of operators Djand D−j a,j∈N0. In fact, more general differential and antidifferential operators are strongly omnipresent, see [6, Sections 3–4], [8] and Section 4. In [9] sufficient conditions are given for an operator to be strongly omnipresent, as well as characterizations of the strong omnipresence of composition and multiplication operators. Finally, we will need in Section 3 some terminology taken from the modern theory of universality. The reader is referred to [19] for an excellent survey about the history, results and references on this topic. If Xand Yare (Hausdorff) topological vector spaces over the same field K(= Ror C) and Tn:X→Y(n∈N) is a sequence of continuous linear mappings, then (Tn) is said to be hypercyclic (or universal) whenever there is a vector x∈X, called also hypercyclic for (Tn), such that the orbit {Tnx:n∈N}is dense in Y. Note that this forces Yto be separable. The sequence (Tn) is called densely hypercyclic whenever the set HC((Tn)) of hypercyclic vectors for (Tn) is dense. On the other hand, (Tn) is said to be hereditarily hypercyclic whenever (Tnk) is hypercyclic for each sequence n1< n2< n3<· · · of positive integers. The sequence (Tn) is densely hereditarily hypercyclic if and only if (Tnk) is densely hypercyclic for every sequence n1< n2< n3<· · · as above. For the sake of convenience, we will keep all these definitions even in the case that the mappings Tnare not linear. Finally, if M⊂Xis a linear manifold then we say that it is hypercyclic for (Tn) whenever M\ {0} ⊂ HC((Tn)). In [4, Theorem 2] the following result is obtained. 5 Theorem 1.1. Let Xand Ybe two metrizable topological vector spaces such that Xis separable. Assume that Tn:X→Y(n∈N)is a densely hereditarily hypercyclic sequence of continuous linear mappings. Then there is a dense linear submanifold of Xall of whose non-zero vectors are hypercyclic for (Tn). Applications of the latter theorem can be found in [4, Theorems 3–4] and [20]. In fact, Theorem 1.1 is an extension of the known result of Herrero–Bourdon–B`es asserting the existence of T–invariant dense hypercyclic linear manifolds for a hypercyclic linear operator T(i.e, the sequence of iterates (Tn) is hypercyclic) on a (real or complex) locally convex space, see [10, 11, 21] (see also [5] to add the property “with maximal cardinality” to such manifolds when Tacts on a Banach space). 2 Totally omnipresent operators In this section we first define in a practical way a new kind of operator. We then show how that definition can be translated in terms of approximation of vectors in certain function spaces. Let us denote by N(∂G) the family of all sequences of similarities of the plane which take the unit disk near the boundary of G, that is, N(∂G) = {σ= (τn)⊂LT (G) : τnis non–constant (n∈N) and supz∈Dχ(τn(z), ∂G)→0 (n→ ∞)}, where χdenotes the chordal distance on C∞. Observe that since ∂G is compact in C∞the fact (τn)∈N(∂G) implies the existence of at least one boundary point tand of a sequence {n1< n2< n3<· · ·} ⊂ Nwith τnk→t(k→ ∞) uniformly on D. If Tis an operator on H(G), g∈H(D), ε > 0, r∈(0,1) and σ= (τn)∈N(∂G) then we set U?(T, g, ε, r, σ) = {f∈H(G) : there is n∈Nwith k(Tf)◦τn−gkrD< ε}.(1) Definition 2.1. Let T:H(G)→H(G) be an operator. We say that Tis totally omnipresent whenever each set U?(T, g, ε, r, σ) is dense in H(G) (g∈H(D), ε > 0, r∈(0,1), σ∈N(∂G)). Note that each U?(T, g, ε, r, σ) is an open set of H(G). For future references, we denote by D(h, K, δ) (h∈H(G), δ > 0, Ka compact subset of G) the basic neighborhood D(h, K, δ) = {f∈H(G) : kf−hkK< δ}.(2) 6 Remark 2.1. If (gi)is a dense sequence in H(D)(for instance, (gi)may be an enumeration of polynomials with coefficients having rational real and imaginary parts) then Tis totally omnipresent if and only if for each σ∈N(∂G)and each (i, j)∈N2the set U?(T, gi,1 j,j j+1, σ) is dense in H(G). As promised, we reformulate the last definition in other language. Before this, a little more notation: If t∈∂G then N(t) will stand for the set of all sequences (τn) of non– constant affine linear mappings with τn(D)⊂G(n∈N) and τn(z)→t(n→ ∞) uniformly on D. Trivially, N(t)⊂N(∂G). On the other hand, Cτdenotes composition with the function τ(i.e., Cτ(h) = h◦τ) whenever it makes sense. In the next proposition, the equivalence (b)⇐⇒(c) is trivial, but we want to establish (c) explicitely because the implication (a)=⇒(c) will be crucial in the proof of Theorem 5.1. Proposition 2.2. Let Tbe an operator on H(G). Then the following conditions are equivalent: (a) The operator Tis totally omnipresent. (b) For every t∈∂G and every (τn)∈N(t)there exists a dense set of functions f∈H(G) satisfying that for every g∈H(D)there exists a strictly increasing sequence (nk)⊂N such that (Tf)(τnk(z)) →g(z) (k→ ∞)uniformly on compact subsets of D. In other words, the sequence Cτn◦T:H(G)→H(D) (n∈N)is densely hypercyclic. (c) For every t∈∂G and every (τn)∈N(t), the sequence Cτn◦T:H(G)→H(D) (n∈N) is densely hereditarily hypercyclic. Proof. Let (gi) be a countable dense set in H(D). Given t∈∂G and σ= (τn)∈N(t), the set M(σ) := \ i,j∈N U?(T, gi,1 j,j j+ 1, σ) is the set of hypercyclic vectors for {Cτn◦T}n≥1. So (a)⇐⇒(b) follows from the fact that H(G) is a Baire space. We now consider the relationship between total and strong omnipresence. If we consider a set U(T, g, ε, r, V ) as in Definition 1.1(c) then we can associate to Va point t∈V∩(∂G) as well as a sequence of open balls Bn⊂G∩V(n∈N) such that supw∈Bnχ(w, t)→0 (n→ ∞). Then sup z∈D χ(τn(z), ∂G)≤sup z∈D χ(τn(z), t) = sup w∈Bn χ(w, t)→0 (n→ ∞), 7 where τn(z) is a non-constant affine linear mapping with τn(D) = Bn. Therefore σ:= (τn)∈N(∂G). If Tis totally omnipresent then U?(T, g, ε, r, σ) is dense in H(G). If f∈U?(T, g, ε, r, σ) then there exists N∈Nsuch that k(Tf)◦τN−gkrD< ε, so f∈ U(T, g, ε, r, V ) because τN∈LT(V) since τN(D) = BN⊂G∩V. Summarizing, U?(T, g, ε, r, σ)⊂U(T, g, ε, r, V ). Thus, the last set is dense. Hence we have proved that every totally omnipresent operator is strongly omnipresent. In Section 4 we will see several examples of (linear) strongly omnipresent operators (in fact, composition operators) which are not totally omnipresent. Further examples will be provided at the end of Section 5 and after Theorem 6.1. 3 Common hypercyclic linear manifolds In this section we are going to improve Theorem 1.1 in order to use that improvement in Section 5. Observe that the next result asserts the existence of common large hypercyclic manifolds for a countable family of sequences of linear mappings. It should be pointed out that the unique additional hypothesis with respect to Theorem 1.1 is that Xis Baire, which takes place, for instance, if Xis complete. Theorem 3.1. Let Xand Ybe two metrizable topological vector spaces such that Xis Baire and separable. Assume that, for each k∈N,T(k) n:X→Y(n∈N)is a densely hereditarily hypercyclic sequence of continuous linear mappings. Then there is a dense linear submanifold M⊂Xsuch that M\ {0} ⊂ \ k∈N HC((T(k) n)). Proof. Observe first that hypercyclicity forces Yto be separable, so second–countable. Let us choose a dense sequence (zn) in Xand denote by da distance on Xcompatible with its topology. We will consider later the open balls GN={x∈X:d(x, zN)<1 N}(N∈N). Since Xis a Baire space and Yis second–countable each of the sets HC((T(k) n)) (k∈N) is residual in X[19, Theorem 1], because they are dense. Therefore their intersection Tk∈NHC((T(k) n)) is also residual, so dense, whence we can pick a vector x1∈G1∩\ k∈N HC((T(k) n)). 8 Then for every k∈Nwe can find a (strictly increasing) subsequence {p(1, k, j) : j∈N}of positive integers such that T(k) p(1,k,j)x1→0 (j→ ∞). But, since each (T(k) n) (k∈N) is densely hereditarily hypercyclic, every set HC((T(k) p(1,k,j))) is again residual. Thus, as above, a vector x2can be selected in G2∩Tk∈NHC((T(k) p(1,k,j))). Now choose for every ka subsequence {p(2, k, j) : j∈N}of (p(1, k, j)) with T(k) p(2,k,j)x2→0 (j→ ∞). Note that also T(k) p(2,k,j)x1→0 (j→ ∞) for each k∈N. Since the new sequences (T(k) p(2,k,j)) (k∈N) are again densely hypercyclic, one can choose a vector x3∈G3∩ Tk∈NHC((T(k) p(2,k,j))). It is evident that this process can be continued by induction, getting a sequence {xN: N∈N} ⊂ Xand a family {{p(n, k, j) : j∈N}:n, k ∈N}of sequences of positive integers satisfying xN∈GNfor all N∈N,(3) xN∈\ k∈N HC((T(k) p(N−1,k,j))) for all N∈N(4) and T(k) p(n,k,j)xN→0 (j→ ∞) for all n≥Nand all k∈N,(5) where, in order to make the notation consistent, (p(0, k, j)) stands for the whole sequence of positive integers for every k∈N. Define M= span({xN:N∈N}). Since {zn:n∈N}is dense in Xand d(xn, zn)<1 n→0 (n→ ∞) (by (3)), the set {xn:n∈N}is also dense, hence Mis a dense linear submanifold of X. It remains to prove that each nonzero vector of Mis hypercyclic for each sequence (T(k) n) (k∈N). Fix x∈M\{0}. Then there are finitely many scalars a1, . . . , aNwith aN6= 0 such that x=PN n=1 anxn. Since a nonzero multiple of a hypercyclic vector is still hypercyclic, we may assume that aN= 1. Fix a positive integer kand a vector y∈Y. Let us show a subsequence {T(k) r(j):j∈N}of (T(k) n) such that T(k) r(j)x→y(j→ ∞). 9 Remark 4.6. A closer look at the last proof (a suitable subsequence of (τn)tending to some boundary point will be needed) reveals that in order that Tbe totally omnipresent it suffices that the following property holds: For each compact subset K⊂Gand each t∈∂G there is an open set Vwith V3tsuch that, for every closed ball B⊂V∩G, (i) The restriction mapping TB0has dense range and (ii) For each f∈H(G)and ε > 0there exist a compact set S⊂G\Kwith connected complement and δ > 0such that for all g∈H(G)the fact kf−gkS< δ implies kTf −TgkB< ε. Note that if Tis linear then (ii) reduces to say (ii’) For each ε > 0there exist a compact set S⊂G\Kwith connected complement and δ > 0such that if g∈H(G)and kgkS< δ then kT gkB< ε. Only a piece of caution: the weaker notions of “somewhere local stability” and “somewhere local density” introduced in [9] do not work here, because they do not permit to fix a sequence (τn) tending to a given boundary point. As a consequence of Theorem 4.5 we obtain that, in particular, if Tis an onto locally stable operator (not necessarily linear) then it is totally omnipresent, compare with Remark 4.2. In addition, we derive again, independently, that the identity operator is totally omnipresent. We are now passing to study the total omnipresence of the right–composition operator Cϕgenerated by a holomorphic selfmapping ϕ∈H(G, G). Its strong omnipresence has been recently characterized in [9, Theorem 3.1]. Specifically, it is proved there that Cϕis strongly omnipresent if and only if M(Cϕ) is non-empty if and only if for every V∈O(∂G) the set ϕ(V∩G) is not relatively compact in G. In particular, if G=C, then Cϕis strongly omnipresent if and only if ϕis non–constant. Unfortunately, we have not been able this time to isolate the exact conditions for Cϕto be totally omnipresent, see Theorems 4.8–4.9 below. At this point it is convenient to introduce a new concept and to recall a topological notion. Definition 4.1. We say that a function F:G→Cis locally one–to–one near the boundary if and only if there is a compact set K⊂Gsuch that Fis one–to–one on every open ball U⊂G\K. By using the compactness of ∂G in C∞, it is easy to see that Fis locally one–to–one near the boundary if and only if we can associate to each t∈∂G an open set V⊂C∞ 16 containing tsatisfying that Fis one–to–one on every open ball U⊂V∩G. A mapping F:X→Ybetween two topological spaces X,Yis called proper if the preimage F−1(K) of each compact subset K⊂Yis compact in X. In our setting, the following lemma will reveal itself to be useful. Lemma 4.7. A continuous selfmapping F:G→Gis proper if and only if for each t∈∂G and every compact set K⊂Gthere exists an open set V⊂C∞with V3tsuch that F(V∩G)∩K=∅. Proof. Assume that Fis proper and that, by the way of contradiction, there exist a boundary point tand a compact subset K⊂Gwith the property that F(Vn∩G)∩K6=∅for all n∈N, where Vnis the chordal ball in C∞with center tand radius 1/n. Then we can select a point zn∈Vn∩Gsuch that F(zn)∈K. Hence F−1(K) is not compact, because (zn)⊂F−1(K)⊂Gbut zn→t∈∂G as n→ ∞. This is a contradiction. Conversely, assume Fis not proper, that is, that there exists a compact subset K⊂G such that F−1(K) is not compact. But, by continuity, F−1(K) is closed in G, so F−1(K) cannot be relatively compact in G. Hence there exist a boundary point tand a sequence (zn)⊂F−1(K) with zn→tas n→ ∞. Given any open set V⊂C∞containing t, we can choose n0∈Nsatisfying zn∈V∩Gfor all n > n0. Therefore F(zn)∈F(V∩G)∩Kfor all n>n0, which tells us that F(V∩G)∩K6=∅. This concludes the proof. In our next theorem we will show how the latter two properties –which are rather practicable– suffice for total omnipresence. In Theorem 4.9 we get at least that the fact that ϕbe proper is necessary, but the local bijectivity should be changed to a kind of (not very pleasant) “(1/3)–local bijectivity near the boundary”, see Theorem 4.9 below. Theorem 4.8. If ϕis proper and locally one–to–one near the boundary then the operator Cϕis totally omnipresent. Proof. We will try to apply Theorem 4.5, or rather Remark 4.6 after it, with T=Cϕ. Hence, our goal is to show that (i) and (ii’) are fulfilled. Fix t∈∂G and a compact set K⊂G. By hypothesis and by Lemma 4.7, there exists an open set V1⊂C∞with V13t such that ϕ(V1∩G)∩K=∅. Since ϕis locally one–to–one near the boundary there is another open set V2with t∈V2⊂V1 satisfying that ϕis one–to–one on every open ball U⊂V2∩G. We show now that V=V2satisfies (i) and (ii’). Fix a closed ball B⊂V2∩G. Then there is an open ball Uwith B⊂U⊂V2∩G. If S:= ϕ(B), then S⊂ϕ(V2∩G)⊂ 17 ϕ(V1∩G)⊂G\K, and C\Sis connected because ϕ:U→ϕ(U) is an isomorphism. It is clear that given ε > 0 and g∈H(G) with kgkS< δ := εthen kCϕgkB=kg◦ϕkB=kgkS< ε. On the other hand, the restriction mapping (Cϕ)B0:f∈H(G)7→ (f◦ϕ)|B0∈H(B0) has dense range, because ϕ:B0→ϕ(B0) is an isomorphism and H(G) is dense in H(ϕ(B0)) by Runge’s theorem (note that ϕ(B0) is a simply connected domain contained in G). Thus, (i) and (ii’) are satisfied, and the proof is concluded. Recall that every totally omnipresent operator is ∂–hypercyclic by Proposition 2.2. Theorem 4.9. Assume that ϕ∈H(G, G)and that Cϕis ∂–hypercyclic. Then ϕis proper and satisfies the following property: For every real number s > 3there exists a compact set K⊂Gsuch that, for every open ball B(a, R)⊂G\K, ϕ is one–to–one on B(a, R/s). Proof. Suppose, by the way of contradiction, that ϕis not proper. Therefore, by Lemma 4.7, there is a boundary point tand a compact set K⊂Gsuch that for each n∈Nwe can select a point zn∈Vn∩Gwith ϕ(zn)∈K, where Vnis the chordal ball with center t and radius 1/n. For every nwe can choose rn>0 such that B(zn, rn)⊂Vn∩G. Define σ= (τn) as τn(z) = rnz+zn. Then τn(D) = B(zn, rn)⊂V∩Gand sup z∈D χ(t, τn(z)) ≤1 n→0 (n→ ∞), therefore σ∈N(t). Consider a function f∈H(G) and the constant function g(z) := 1+M, where M= maxz∈K|f(ϕ(z))|. Then g∈H(D) and, for all n∈Nand all r > 0, kCτnCϕf−gkrD≥ |f(ϕ(zn)) −1−M| ≥M+ 1 − |f(ϕ(zn))| ≥ 1. Hence (CτnCϕ) cannot be hypercyclic, which is a contradiction. Assume now, again by the way of contradiction, that ϕdoes not satisfy the (1/3)– property given in the statement. Then there are a real number s > 3 and a sequence of balls B(an, rn)⊂Gtending to the boundary in such a way that ϕis not one–to–one on B(an,rn s). By taking a subsequence if necessary, we can suppose that sup z∈B(an,rn) χ(t, z)→0 (n→ ∞) (11) 18 for some boundary point t. For each positive integer nthere exist points zn, wn∈B(an,rn s) satisfying zn6=wnand ϕ(zn) = ϕ(wn). Consider the following sequence σ= (τn) of affine linear transformations: τn(z) = s 3(wn−zn)z+zn(n∈N). Then τn(0) = zn,τn(3 s) = wnand τn(D) = B(zn,s 3|wn−zn|)⊂B(zn,s 3·2rn s)⊂B(an, rn). Consequently, by (11), sup z∈D χ(t, τn(z)) →0 (n→ ∞), that is, σ∈N(t). By hypothesis, there must be a function f∈HC((CτnCϕ)). Thus, for a suitable subsequence (τnj) of (τn), (CτnjCϕf) tends to the identity function g(z) = zin H(D). In particular, f(ϕ(τnj(0))) →0 and f(ϕ(τnj(3/s))) →3/s as j→ ∞. But this would yield that f(ϕ(znj)) →0 and f(ϕ(wnj)) →3/s (j→ ∞), which is a contradiction because both sequences are the same. In the case G=Cthe following corollary is derived from the latter two theorems. Corollary 4.10. Let ϕbe an entire function. We have: (a) If Cϕis ∂–hypercyclic then ϕis a non-constant polynomial. (b) If ϕis a polynomial of degree one or two then Cϕis totally omnipresent. Proof. Due to Picard’s theorem [17, Chapter 9] and to the fact that limz→∞ P(z) = ∞if P is a non-constant polynomial, only these polynomials are proper, hence the transcendental entire functions are excluded from ∂–hypercyclicity by Theorem 4.9. This proves (a). As for (b), if ϕis a polynomial of degree one, then ϕis bijective from Conto C, so it is locally one–to–one near the boundary and Theorem 4.8 applies. Assume that ϕis a polynomial of degree two, namely, ϕ(z) = az2+bz +c(a, b, c ∈C;a6= 0). Our goal is to get a compact set K⊂Csuch that ϕis one–to–one on every open ball U⊂C\K. Choose K=B(0,1 + |b/a|). Then a ball Uas before would lie on a half–plane Hwhich is at a distance greater than |b/a|from the origin. A simple calculation shows that if ϕ(z) = ϕ(w) and z6=wthen w=−b a−z. But if z∈Uthen z∈H, whence w∈ −b a−H. Hence w6∈ U because H∩(−b a−H) = ∅. Thus, ϕis one–to–one on U, as required. By Theorem 4.9 and Corollary 4.10 we can furnish a new (even linear) example of a strongly omnipresent operator which it is not totally omnipresent. 19 Example 4.11. Choose G=Dand let ϕbe the Blaschke product with zeros at the points zn= 1 −1 n2(n∈N), that is, ϕ(z) = ∞ Y n=1 n2z−n2+ 1 (n2−1)z−n2(z∈D). Since P(1 − |zn|)<+∞, we have (see [16, II Theorem 6.1]) that ϕ∈H(D,D) and that ϕextends to a continuous function on D\ {1}with |ϕ(z)|= 1 on (∂D)\ {1}. Then (∂D)∩∂ϕ(V∩G) is not empty for all V∈O(∂D), hence ϕ(V∩G) is not relatively compact in D, so Cϕis strongly omnipresent. However, Cϕis not totally omnipresent because ϕis not proper, since, for instance, ϕ−1({0}) = {zn:n∈N}, which is not compact. In the case G=Cany Cϕwith ϕtranscendental is strongly omnipresent but not totally omnipresent. As for the case when ϕis a polynomial, we raise the following. Conjecture 4.12. Let ϕbe a polynomial of degree 3 or larger. Then Cϕis not totally omnipresent. We shall be content for now by proving that if Nis a positive integer with N≥10 and ϕ(z) = zNthen Cϕis not totally omnipresent. According to Theorem 4.9, this will be achieved as soon as we can show a real number s > 3 in such a way that to each r > 0 we can associate a ball B(a, R)⊂ {|z|> r}with the property that ϕis not one– to–one on B(a, R/s). Since sin π 5<0.31, we have that sin 2π N≤sin π 5<1 3. Choose swith sin π N<1 s<1 3, and fix r > 0. Since R/s R+r→1 sas R→ ∞, we can select an R > 0 with sin π N<R/s R+r. Consider the balls B(a:= R+r, R), and B(R+r, R/s). The first one is in {|z|> r}while the second one is tangent to two rays from the origin making an angle of opening 2 arcsin R/s R+r, which is greater than 2π/N. But given w0∈C\ {0}the roots of ϕ(z) = w0are Npoints in the circle |z|=|w0|1/N equally distributed with angular distance equal to 2π/N. Then for a suitable w0there are at least two roots of ϕ(z) = w0in the second ball, that is, ϕis not one–to–one on B(a, R/s). We are now assuming that Lψis the left–composition operator on H(G) associated to an entire function ψ. In [9, Section 3] it is asserted that Lψis strongly omnipresent on H(G) if and only if M(Lψ)6=∅if and only if ψhas an approximate right inverse, that is, there is a sequence (fn) of entire functions such that ψ(fn(z)) →z(n→ ∞) locally uniformly in C. The proof is based there on the fact that locally dense range plus local stability near the boundary imply strong omnipresence. But they also imply total omnipresence, see Theorem 4.5. Consequently, we are allowed to establish the following theorem. Theorem 4.13. Let Lψbe the left–composition operator on H(G)defined by ψ∈H(C). Then the following assertions are equivalent: 20 (a) The operator Lψis totally omnipresent. (b) The operator Lψis strongly omnipresent. (c) The operator Lψis ∂–hypercyclic. (d) The set M(Lψ)is non–empty. (e) The function ψhas an approximate right inverse. In particular, ψmust be surjective in order that Lψbe totally omnipresent; but surjectivity alone is not sufficient (see [9]). We finish this section by considering the multiplication operator Mhgenerated by a function h∈H(G). In [9] it is shown that Mhis strongly omnipresent if and only if his non-zero. It is easy to realize that a stronger condition is needed for total omnipresence. Theorem 4.14. Assume that h∈H(G). Then the following are equivalent: (a) The operator Mhis totally omnipresent. (b) The operator Mhis ∂–hypercyclic. (c) The set of zeros of his finite. Proof. That (a) implies (b) is due to Proposition 2.2. Suppose now that (b) holds and that the set of zeros of his not finite. Then the Analytic Continuation Principle allows to assume the existence of a sequence (zn)⊂Gtending to some boundary point tsuch that h(zn) = 0 for all n. For each n, let us choose rn>0 so small that Bn:= B(zn, rn)⊂Gand supz∈Bnχ(t, Bn)→0 (n→ ∞). Define the mappings τn(z) := rnz+zn. Then (τn)∈N(t), whence there exists f∈H(G) such that the sequence {h(τn(z))f(τn(z)) : n∈N}is dense in H(D), which is not possible, because h(τn(0))f(τn(0)) = 0 for all n. This contradiction shows that (b) implies (c). Finally, assume that the hypothesis of (c) is fulfilled, that is, h(z)6= 0 for all z∈G\K, for some compact set K⊂G. If U⊂G\Kis an open ball then the operator f∈H(G)7→ (h·f)|U∈H(U) has dense range by Runge’s theorem. Hence Mh has locally dense range near the boundary. Moreover, Mhis obviously locally stable near the boundary (for any h). An application of Theorem 4.5 yields that (c) implies (a). Observe that from the last theorem we obtain further examples of linear strongly non– totally omnipresent operators: just take G=Dand T=Mh, where his the Blaschke product ϕconsidered in Example 4.11. 21 5 Dense linear manifolds of monsters The content of this section has been the main motivation for this paper. As indicated in Section 1, Luh [22] and Grosse–Erdmann [18] showed that, topologically speaking, the set of Luh–monsters is huge. We prove here that not only topologically but also algebraically Luh “created” too many monsters. The precise formulation for this statement will be given in Theorem 5.2. Nevertheless, a more general result can be stated. Theorem 5.1. Assume that (Sj)is a countable family of linear totally omnipresent operators on H(G). Then there exists a dense linear submanifold M⊂H(G)such that M(Sj)⊃M\ {0}for all j. Proof. Fix a dense sequence {tk:k∈N}in ∂G. For each k∈N, fix a sequence (τ(k) n)∈ N(tk). By Proposition 2.2, the sequence T(k,j) n:X→Y(n∈N) is densely hereditarily hypercyclic for every k∈Nand every j, where X=Y:= H(G) and T(k,j) n:= Cτ(k) nSj. Then the hypotheses of Theorem 3.1 are fulfilled. Hence there is a dense linear manifold M⊂H(G) with M\ {0} ⊂ \ k,j HC((Cτ(k) nSj)). But observe that the last intersection is included in M(Sj) for each j, because in order that Sjfbe a holomorphic monster it is sufficient to see its wild behaviour only near the points of a dense boundary subset, see [6, Lemma 2.1]. This drives us to M\ {0}⊂M(Sj) for all j. The last theorem yields immediately the next corollary. Theorem 5.2. Assume that G⊂Cis a simply connected domain. Then there exists a dense linear submanifold of H(G)whose non–zero members are Luh–monsters. Proof. Fix a point a∈Gand define the operators Sj(j∈Z) as Sj=Dj(j∈N0), Sj=Dj a (−j∈N). Now just apply Theorem 5.1 and take in mind that {Luh–monsters}=\ j∈Z M(Sj). Of course, Theorem 5.1 holds when the sequence (Sj) is changed by just a single operator Ton H(G). One can believe that the same assertion of Theorem 5.1 would hold just by 22 assuming that Tis a linear strongly omnipresent operator. This is false. As a matter of fact, it can happen that M(T) does not contain any manifolds of dimension greater than one. Indeed, consider the linear operator Tf(z) = f(0)·ϕ(z) on H(D), where ϕ∈H(D) is a fixed holomorphic monster (see [9, Example 2.9]). Assume that Mis a linear manifold with M\ {0}⊂M(T) and dim(M)≥2. Hence we can select two linearly independent functions f,gin M. Since M(T) = {h∈H(D) : h(0) 6= 0}we have α:= f(0) 6= 0 6=g(0) =: β. Define h(z) := f(z)−α βg(z). By linear independence, h∈M\ {0}, whence h∈ M(T). Thus, 0 6=h(0) = α−α= 0, which is a contradiction. This shows that dim(M)≤1, as required. By the way, this shows that the operator Tis not totally omnipresent. 6 Relationship to the DI–operators. The Volterra operator In this section we are considering briefly the boundary wild behaviour from another point of view. In 1995 the first author proved [2] that for every subset A⊂Gwhich is not relatively compact in Gthere exists a residual set Min H(G) such that f(n)(A) is dense in Cfor every n∈N0, and the second author showed later [12] that the same property is shared by certain kinds of infinite order differential and antidifferential operators and Volterra operators. This motivated us [7] to introduce the notion of dense–image operators or, briefly, DI–operators. A DI–operator is a (not necessarily linear) continuous selfmapping Ton H(G) satisfying that the set M(T, A) := {f∈H(G) : T f(A) is dense in C}is dense in H(G) for every subset A⊂Gwhich is not relatively compact in G. From the fact that any of these subsets contains a sequence tending to some boundary point, it is easy to see that every totally omnipresent operator is a DI–operator. The converse is not true, see below. Let us summarize in one theorem several examples of relevant classes of DI–operators, see [7] and [12]. The interested reader should compare the following to the results of Section 4 and note that these earlier results improve in part the content of the next theorem. Theorem 6.1. Let G⊂Cbe a domain, Φ(z),Ψ(z)two power series as in Theorem 4.1, and let ϕ:G×G→Cbe a holomorphic function with respect to both variables. Assume also that ϕ1∈H(G, G),ϕ2∈H(C),ϕ3∈H(G). We have: (a) If Φis non–zero then Φ(D)is a DI–operator. (b) If Gis simply connected, a∈Gand Vϕis the Volterra operator associated to a,ϕ, then Vϕis a DI–operator if and only if for every compact subset L⊂Gand every A⊂Gwhich is not relatively compact there exist b∈A\Land s∈Gsuch that ϕ(b, s)6= 0. In addition, if ϕsatisfies this property then Φ(D) + Vϕis a DI–operator. In particular, if at least one of Φ,Ψis non–zero then Φ(D)+Ψ(D−1 a)is a DI–operator. (c) Every onto linear operator is a DI–operator. 23 (d) The operator Cϕ1is DI if and only if ϕ1is proper. In particular, if G=C, then Cϕ1 is a DI-operator if and only if ϕ1is a non–constant polynomial. (e) The operator Lϕ2is DI if and only if ϕ2is non–constant. (f) The operator Mϕ3is DI if and only if the set of zeros of ϕ3is finite. In [7] it is proved that DI property implies omnipresence, and linear examples are exhibited showing that strong omnipresence (so omnipresence) does not imply DI–property. A non–linear example of a DI–operator which is not strongly omnipresent is also furnished, but we do not know whether every linear DI–operator is strongly omnipresent. If G=C and ϕ(z) = z10 then by part (d) of the latter theorem the linear operator Cϕis DI (it is also strongly omnipresent, because ϕis not constant), but it is not totally omnipresent, as we saw after Conjecture 4.12. Moreover, if G=C,a= 0 and ϕ(z, t) := sin(πz) then the linear operator Vϕis strongly omnipresent by [8] but it is not DI (so not totally omnipresent): choose L=∅and A=Nin part (b) of Theorem 6.1. Now, we focus our attention on the Volterra operator of the first kind Vϕand conclude this paper with the statement of two results whose contents and proofs are analogous to the corresponding ones in [8]. Hence their proofs are left to the interested reader. In fact, similarly to [8], the first result below can be used to prove the second one as well as some assertions of Theorem 4.1. In the following, Gis a simply connected domain of Cand ais a fixed point in G. In addition, if Bis a closed ball in G, then A(B) denotes the Banach space of all functions that are continuous in Band holomorphic in B0, endowed with the maximum norm k · kB. With the same norm we endow the subspace Ab(B) consisting of all functions of A(B) with a zero at b, where b∈∂B. Theorem 6.2. Let S:H(G)→H(G)be an operator and ϕ:G×G→Ca holomorphic function with respect to both variables. Then the operator on H(G)defined by Tf(z) = Sf(z) + Vϕ(z) is totally omnipresent if there exists a compact set K⊂Gsuch that for each closed ball B0⊂G\Kthere is a closed ball Bwith B0⊂B⊂G\Kand a point b∈∂B such that (a) the operator Sextends continuously to a mapping e S:A(B)→A(B0), (b) the mapping e T:Ab(B)→A(B0)defined by e Tf(z) = e Sf(z) + Zz b f(t)ϕ(z, t)dt (z∈B0) has dense range. 24 Theorem 6.3. Assume that ϕ:G×G→Cis holomorphic and that there exist N∈N0 and a compact set K⊂Gsuch that ∂Nϕ ∂zN(w, w)6=0=∂nϕ ∂zn(w, w) (n= 0,1, . . . , N −1) for all w∈G\K. Then the operator Vϕis totally omnipresent. We propose here the problem of characterizing the total omnipresence of Vϕin terms of ϕ. ACKNOWLEDGEMENT The authors are grateful to the referees for helpful comments and suggestions. References [1] L. Bernal–Gonz´alez, Omnipresent holomorphic operators and maximal cluster sets, Colloq. Math. 63 (1992), 315–322. [2] L. Bernal–Gonz´alez, Plane sets having dense holomorphic images, Rev. Roumaine Math. Pures Appl. 40 (1995), 567–569. [3] L. Bernal–Gonz´alez, Hypercyclic sequences of differential and antidifferential operators, J. Approx. Theory 96 (1999), 323–337. [4] L. Bernal–Gonz´alez, Densely hereditarily hypercyclic sequences and large hypercyclic manifolds, Proc. Amer. Math. Soc. 127 (1999), 3279–3285. [5] L. Bernal–Gonz´alez, Universal images of universal elements, Studia Math. 138 (2000), 241–250. [6] L. Bernal–Gonz´alez and M.C. Calder´on–Moreno, Holomorphic T–monsters and strongly omnipresent operators, J. Approx. Theory 104 (2000), 204–219. [7] L. Bernal–Gonz´alez and M.C. Calder´on–Moreno, Operators with dense images everywhere, J. Math. Anal. Appl. 263 (2001), 95–109. [8] L. Bernal–Gonz´alez, M.C. Calder´on–Moreno and K.G. Grosse–Erdmann, Strongly omnipresent integral operators, Integral Equ. Oper. Th., in press. 25