Subspaces of frequently hypercyclic functions for sequences of

SUBSPACES OF FREQUENTLY HYPERCYCLIC FUNCTIONS FOR SEQUENCES OF COMPOSITION OPERATORS

arXiv:1712.06339v1 [math.CV] 18 Dec 2017

´ ´ L. BERNAL-GONZALEZ, M.C. CALDERON-MORENO, A. JUNG, AND J.A. PRADO-BASSAS Abstract. In this paper, a criterion for a sequence of composition operators defined on the space of holomorphic functions in a complex domain to be frequently hypercyclic is provided. Such criterion improves some already known special cases and, in addition, it is also valid to provide dense vector subspaces as well as large closed ones consisting entirely, except for zero, of functions that are frequently hypercyclic.

1. Introduction The phenomenon of hypercyclicity has become a trend in the last three decades. Roughly speaking, it means density of some orbit of a vector under the action of an operator or a sequence of operators. When such density is quantified in some optimal sense, the property of frequent hypercyclicity – a concept coined by Bayart and Grivaux [4] – arises naturally. In this paper we are concerned with the study of frequent hypercyclicity of sequences of composition operators acting on holomorphic functions defined in a domain of the complex plane. But prior to going on, let us fix some notation and definitions, mostly standard. By N, R, C, D, D and T we denote, respectively, the set of positive integers, the real line, the complex plane, the open unit disc {z ∈ C : |z| < 1}, the closed unit disc {z : |z| ≤ 1} and the unit circle {z ∈ C : |z| = 1}. A domain is a nonempty connected open subset G ⊂ C. Its one-point compactification G ∪ {∞} will be represented by G∞ . A domain G ⊂ C is called simply connected provided that C∞ \ G is connected. The symbol H(G) will stand for the space of holomorphic functions on G. It becomes an F-space – that is, a complete metrizable topological vector space – under the topology of uniform convergence on compact subsets of G. Next, we recall the notion of hypercyclicity. For a good account of concepts, results and history concerning this topic, the reader is referred to the books [5, 29]. Let X and Y be two (Hausdorff) topological vector spaces. Then L(X, Y ) represents the family of all linear continuous mappings X → Y , while L(X) denotes the 2010 Mathematics Subject Classification. 30E10, 47B33, 47A16, 47B38. Key words and phrases. hypercyclic sequence of operators, composition operator, frequent hypercyclicity, holomorphic function, lineability. 1

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´ BERNAL, CALDERON, JUNG, AND PRADO

set of operators on X, that is, L(X) = L(X, X). A sequence (Tn )n ⊂ L(X, Y ) is said to be hypercyclic provided that there is a vector x0 ∈ X (called hypercyclic for (Tn )n ) such that the orbit {Tn x0 : n ∈ N} is dense in Y . An operator T ∈ L(X) is said to be hypercyclic if the sequence (T [n] )n of its iterates (i.e. T [1] := T and T [n+1] := T ◦ T [n] ) is hypercyclic. The corresponding sets of hypercyclic vectors will be respectively denoted by HC((Tn )n ) and HC(T ). If X is an F-space and T ∈ L(X) is hypercyclic, then HC(T ) is residual (in particular, dense) in X. If X is an F-space, Y is metrizable and separable and (Tn )n ⊂ L(X, Y ), then HC((Tn )n ) is residual if and only if it is dense, and if and only if (Tn )n is transitive (meaning that, given nonempty open sets U ⊂ X, V ⊂ Y , there is N ∈ N such that TN (U) ∩ V 6= ∅). As a stronger notion of hypercyclicity, the property of frequent hypercyclicity was introduced in [4] for operators, and then extended in [19] to sequences of linear mappings. Given a subset A ⊂ N, the lower density of A is defined as

card (A ∩ {1, . . . , n}) . n→∞ n Assume that X and Y are topological vector spaces. Then a sequence (Tn )n ⊂ L(X, Y ) is called frequently hypercyclic provided that there is a vector x0 ∈ X (called frequently hypercyclic for (Tn )n ) such that dens (A) = lim inf

dens ({n ∈ N : Tn x0 ∈ V }) > 0 for every nonempty open set V ⊂ Y . An operator T ∈ L(X) is said to be frequently hypercyclic if the sequence (T [n] )n of its iterates is frequently hypercyclic. The corresponding sets of frequently hypercyclic vectors will be respectively denoted by F HC((Tn )n ) and F HC(T ). The set F HC(T ) is dense as soon as it is nonempty but, in contrast to mere hypercyclicity, it is always of the first Baire category if X is an F-space, as proved in [26, Prop. 4.9 and Remark 4.10] (see also [6,34] and [22,27] for other degrees of hypercyclicity). If X is an F-space and (Tn )n ⊂ L(X), then we can proceed as in [29, Proposition 9.19] (which was the special case Tn = T [n]) so as to assert that if Tn → 0 pointwise on some dense subset of X (this condition is fulfilled if, for instance, (Tn )n satisfies the so-called Frequent Hypercyclicity Criterion, see [19, 20]), then F HC((Tn )n ) is of the first Baire category. Our main concern enters the realm of composition operators. If G ⊂ C is a domain, then every holomorphic selfmap ϕ : G → G defines a composition operator Cϕ ∈ L(H(G)) given by Cϕ : f ∈ H(G) 7−→ f ◦ ϕ ∈ H(G). By H(G, G), H1−1 (G) and Aut(G) we denote, respectively, the family of such selfmaps, the subfamily of all univalent (i.e., one-to-one) members of H(G, G), and the subfamily of all automorphisms of G (that is, the bijective holomorphic functions G → G). Hence Aut(G) ⊂ H1−1 (G) ⊂ H(G, G). According to [12],

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a sequence (ϕn )n ⊂ H(G, G) is said to be runaway whenever it satisfies the following property: for every compact set K ⊂ G, there exists N ∈ N such that K ∩ ϕN (K) = ∅.

A function ϕ ∈ H(G, G) is said to be runaway if the sequence (ϕ[n] ) of its iterates is runaway. In [3, 4, 12, 17, 28, 38, 39] the dynamics of a sequence (Cϕn )n is studied, so as to obtain, among others, the following assertions, in which G is assumed to be a simply connected domain and ϕ ∈ H(G, G): • The operator Cϕ is hypercyclic if and only if ϕ ∈ H1−1 (G) and has no fixed point in G, and if and only if ϕ ∈ H1−1 (G) and is runaway. • Let (ϕn )n := (z
7→ an z + bn )n ⊂ Aut(C) (an , bn ∈ C; an 6= 0) and (ψn )n := z−an z 7→ kn 1−an z n ⊂ Aut(D) (|an | < 1 = |kn |). Then (Cϕn )n is hypercyclic if and only if lim supn→∞ min{|bn |, |bn /an |} = +∞, and (Cψn )n is hypercyclic if and only if lim supn→∞ |an | = 1. Birkhoff’s theorem [18] for G = C is the special case an = 0, bn = an (a 6= 0). • Let (ϕn )n ⊂ H1−1 (G). Then (Cϕn )n is hypercyclic if and only if (ϕn )n is runaway. • Each translation τa : f ∈ H(C) 7→ f (· + a) ∈ H(C) (a ∈ C \ {0}) is frequently hypercyclic. If ϕ ∈ Aut(D) and it has no fixed point in D, then Cϕ : H(D) → H(D) is also frequently hypercyclic. • The operator Cϕ is frequently hypercyclic if and only if it is hypercyclic, so if and only if ϕ ∈ H1−1 (G) and has no fixed point in G. • Let (ϕn )n := (z 7→ an z + bn )n ⊂ Aut(C) (an , bn ∈ C; an 6= 0). Assume that there exists an unbounded nondecreasing sequence (ωn )n ⊂ (0, +∞) satisfying lim (|bn | − ωn |an |) = +∞ and n→∞

|bm − bn | ≥ ωm−n (|am | + |an |) for all m, n ∈ N with m > n.

Then (Cϕn )n is frequently hypercyclic on H(C). • If (bn )n ⊂ C is a sequence such that lim inf |bn+k − bn | = +∞,

k→∞ n∈N

then the sequence of translations (τbn )n is frequently hypercyclic on H(C). The aim of this paper is to furnish sufficient conditions for a sequence of composition operators defined on a given planar simply connected domain to be frequently hypercyclic and to enjoy the property that the family of its corresponding frequently hypercyclic functions is not only nonempty but also it contains vector spaces that are, in several senses, large. This will be performed in Sections 3 and 4. Section 2 will be devoted to provide further necessary terminology together with a number of useful assertions and several results motivating our research. In the final Section 5 a number of examples will be provided.

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2. Lineability terminology, lineability within hypercyclicity, and some auxiliary results That a subset A of a complete metric space is residual means that, in a topological sense, A is large. Recently, a new terminology has been coined with the aim of suggesting largeness in an algebraic sense. Such terminology, part of which is sketched in the next paragraph, enters the new subject of lineability, for whose background we refer the reader to the survey [15] and the book [2]. Assume that X is a vector space and that A ⊂ X. The set A is said to be lineable provided that there is an infinite dimensional vector space M such that M ⊂ A ∪ {0}. If X is, in addition, a topological vector space, then A is called: dense-lineable in X if there exists a dense vector subspace M ⊂ X such that M ⊂ A ∪ {0}; maximal dense-lineable in X if, further, dim(M) = dim(X); spaceable in X if there exists a closed infinite dimensional vector subspace M ⊂ X such that M ⊂ A ∪ {0}.

Let X, Y be topological vector spaces, T ∈ L(X) and (Tn )n ⊂ L(X, Y ). Turning to our setting of hypercyclicity and using the preceding language, it is known (see [16, 23, 31, 40]) that if T is hypercyclic, then HC(T ) is always dense-lineable in X, even maximal dense-lineable if X is a Banach space (with the additional property, in both cases, that the existing dense subspace M is T -invariant). Sufficient conditions for HC((Tn )n ) to be lineable or dense-lineable were furnished in [8, Theorems 1–2] (for maximal dense lineability of HC((Tn )n ), see [14]). In contrast to the property of dense-lineability, not every set HC(T ) is spaceable if T is hypercyclic, as Montes shown in [33]. Sufficient criteria for HC(T ) or HC((Tn )n ) to be spaceable can be found in [29, Chapter 10], [2, Section 4.5] and the references contained in them. In particular, if G ⊂ C is a domain that is not conformally equivalent to C\{0} and (ϕn )n ⊂ Aut(G) is a runaway sequence, then HC((Cϕn )n ) is spaceable in H(G) [13]; consequently, HC(Cϕ ) is spaceable if ϕ ∈ Aut(G) is runaway. Assuming that G is simply connected and (ϕn )n ⊂ H1−1 (G) is a runaway sequence, it can be extracted from [28, Theorem 3.2 and its proof] together with [8, Theorem 2] that HC((Cϕn )n ) is dense-lineable in H(G) (see also [11] for additional properties concerning functions having maximal cluster sets along curves tending to the boundary). Concerning frequent hypercyclicity, Bayart and Grivaux [4] proved that if X is a separable F-space and T ∈ L(X) is frequently hypercyclic, then F HC(T ) is dense-lineable (again with T -invariance of the corresponding dense subspace). For conditions for F HC((Tn )n ) to be dense-lineable, see [14]. As for the existence of large closed subspaces, in [21, Theorem 3] a sufficient criterion for the spaceability of F HC(T ) is given – from which it is derived as an example that F HC(τa ) is spaceable in H(C) for each a ∈ C \ {0} – while Menet [32, Theorem 2.12] provided a criterion to discover spaceability of F HC((Tn )n ). B`es [17] proved in 2013 the

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spaceability in H(G) of F HC(Cϕ ) if it is assumed that G ⊂ C is a simply connected domain, ϕ ∈ H1−1 (G) and ϕ has no fixed point in G.

As far as we know, there is not any general result about lineability of the family of frequent hypercyclic functions with respect to sequences of composition operators. This paper tries to contribute to fill in this gap. To finish this section, we turn our attention to approximation of holomorphic functions. In this context, the next definition and theorem can be found in [25] (see also [35]). Theorem 2.2 below is known as Nersesjan’s approximation theorem. Definition 2.1. Let G ⊂ C be a domain. A relatively closed set F ⊂ G is said to be a Carleman set of G if G∞ \ F is connected and locally connected at ∞ and it “lacks long islands”, that is, for every compact set K ⊂ G there is a neighborhood V of ∞ in G∞ such that no component of F ◦ , the interior of F , meets both K and V . Theorem 2.2. A relatively closed set F ⊂ G is a Carleman set of G if and only if for every function f continuous on F and holomorphic on F ◦ and every positive and continuous function ε : F → [0, +∞), there is a function g ∈ H(G) such that |f (z) − g(z)| < ε(z) for all z ∈ F.

3. Frequently hypercyclic sequences of composition operators Let G ⊂ C be a domain. A sequence (Kn )n of compact subsets of G is said S ◦ to be exhaustive if G = ∞ K n and Kn ⊂ Kn+1 for all n ∈ N. In particular, n=1 it satisfies that, given a compact set K ⊂ G, there exists N ∈ N such that K ⊂ KN . A compact set K ⊂ C is said to be Mergelyan whenever it lacks holes, that is, whenever C \ K is connected. The symbol M(G) will stand for the family of Mergelyan compact subsets of G. If G is simply connected, an exhaustive sequence of compact subsets of G satisfying (Kn )n ⊂ M(G) always exists: see, e.g., [37, Chap. 13]. Given a compact subset K ⊂ G, r > 0 and a function f ∈ H(G), by kf kK we mean the maximum of |f (z)| over K, while BK (f, r) will denote the set of all functions h ∈ H(G) such that kh − f kK < r. The sets BK (f, r) form a basis for the open sets of H(G). We first give an easy necessary condition for a sequence (ϕn )n ⊂ H(G, G) to generate a frequent hypercyclic sequence (Cϕn )n of composition operators. Proposition 3.1. If (Cϕn )n is frequently hypercyclic on H(G), then (ϕn )n is frequently runaway, that is, for every compact set K ⊂ G, one has dens({n ∈ N : K ∩ ϕn (K) = ∅}) > 0.

Proof. Fix any compact set K ⊂ G. Let n ∈ N be a natural number such that K ∩ ϕn (K) 6= ∅ and pick zn ∈ K such that ϕn (zn ) ∈ K. Let f ∈ F HC((Cϕn )n )

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and define g as the constant function g(z) := 1 + kf kK . Then kg − Cϕn f kK ≥ |g(zn ) − f (ϕn (zn ))| ≥ 1 + kf kK − |f (ϕn (zn ))| ≥ 1, where the fact ϕn (zn ) ∈ K ∋ zn has been used. Hence {n ∈ N : Cϕn f ∈ BK (g, 1)} ⊂ {n ∈ N : K ∩ ϕn (K) = ∅}. But the frequent hypercyclicity of f implies that the smaller set has positive lower density, so the bigger one too.  The last proposition can suggest conditions guaranteeing that F HC((Cϕn )n ) 6= ∅. We restrict ourselves to the rather illustrative case of simply connected domains. Neither the results nor the approaches in [17, 21, 32] will be used in our proofs. Lemma 3.2. Any sequence of natural numbers with positive lower density can always be split into infinite many disjoint subsequences each of which has also positive lower density. Proof. Observe that N is the disjoint union of the subsets Zp (p prime), where Z2 is the set of even natural numbers and, for p > 2, Zp is the set of multiplesQof p not included in Zq with q < p. Take also into account that dens (Zp ) ≥ 1/ q≤p q. q prime The result follows from these two facts.  We say that a family F of subsets of a given set X is pairwise disjoint if A ∩ B = ∅ for every pair of distinct A, B ∈ F . If A ⊂ C then ∂A and ∂∞ A will stand for the boundary of A in C and in C∞ , respectively, so that ∂∞ A equals either ∂A or {∞} ∪ ∂A, depending on whether A is bounded or not. Theorem 3.3. Let G ⊂ C be a simply connected domain and (ϕn )n ⊂ H1−1 (G). Assume that there exists an exhaustive sequence (Kν )ν ⊂ M(G) of compact subsets of G as well as a family {A(ν) : ν ∈ N} of subsets of N satisfying the following conditions: (P1) dens (A(ν)) > 0 for all ν ∈ N. (P2) Each of the families {A(ν); ν ∈ N} and K := {ϕn (Kν ) : n ∈ A(ν), ν ∈ N} is pairwise disjoint. (P3) Given a compact subset K ⊂ G there are only finitely many L ∈ K with K ∩ L 6= ∅. Then the sequence (Cϕn )n is frequently hypercyclic. In other words, the set F HC((Cϕn )n ) is not empty. Proof. Let (Pl )l be a dense sequence of H(G), for instance the sequence of polynomials whose coefficients have rational real and imaginary parts. Divide each set A(ν) into infinitely many disjoint subsets A(ν, l) (l ∈ N), each of them with positive lower density. This is possible thanks to (P1) and Lemma 3.2.

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Now, we use the fact that if K is a compact set with N holes and ϕ is an injective holomorphic mapping on a neighborhood of K, then ϕ(K) also has N holes (see [36, p. 276]). In particular, if K ∈ M(G) then ϕ(K) lack holes. Define the set [ [ F := ϕn (Kν ). ν∈N n∈A(ν)

From (P2), the sets ϕn (Kν ) are pairwise disjoint compact subsets of G having no holes. From (P3), it follows that they escape to the boundary of G. Therefore F is closed in G, and G∞ \ F is connected as well as locally connected at ∞. The components of F ◦ are the sets (ϕn (Kν ))◦ . Given a compact subset K ⊂ G there are by (P3) only finitely many L1 , . . . , LN ∈ K with K ∩ L 6= ∅. Then V := {∞} ∪ (G \ (K ∪ L1 ∪ · · · ∪ LN )) is a neighborhood of ∞ in G∞ satisfying that no component of F ◦ meets both K and V . Hence F is a Carleman set of G. Now define g : F → C as follows: ( Pl (ϕ−1 n (z)) if z ∈ ϕn (Kν ), n ∈ A(ν, l), l, ν ∈ N g(z) := 0 otherwise.

It is obvious that g is continuous on F and holomorphic in F ◦ . Hence, by Nersesjan’s Theorem (Theorem 2.2), there exists f ∈ H(G) such that |f (z) − g(z)| < ε(z) for z ∈ F , where ε : G → [0, +∞) is a function that goes to zero as z approaches the boundary of G (for instance, take ε(z) := χ(z, ∂∞ G), where χ denotes the chordal distance on C∞ ). We claim that f is frequently hypercyclic for (Cϕn )n . Indeed, fix l, ν ∈ N and z ∈ Kν . Then, for n ∈ A(l, ν) we have |Cϕn (f )(z) − Pl (z)| = |f (ϕn (z)) − g(ϕn (z))| < ε(ϕn (z)).

(1)

But (P3) implies that ϕn (z) tends to the boundary as n → ∞ uniformly on Kν , so supz∈Kν ε(ϕn (z)) → 0. Hence, given δ > 0, there is n0 = n0 (ν) ∈ N such that supz∈Kν ε(ϕn (z)) < δ for all n ≥ n0 . This, together with (1), gives us that for any l, ν ∈ N, z ∈ Kν and n ∈ A(l, ν) \ {1, . . . , n0 }, we have |Cϕn (f )(z) − Pl (z)| < δ.

Since dens (A(l, ν)\{1, . . . , n0 }) = dens (A(l, ν)) > 0 and the sets BKν (Pl , δ) form a basis for the open sets of H(G), the frequent hypercyclicity of f is proved.  The assumptions of this theorem implies the frequent hypercylicity of the sequence of composition operators. But in the next section we are going to establish, under the same conditions, a large algebraic size of the set F HC((Cϕn )n ). With this in mind, we introduce the next definition. Definition 3.4. Let G ⊂ C be a simply connected domain and (ϕn ) ⊂ H(G, G). We say that (ϕn ) is frequently exhaustively runaway if the conditions of Theorem 3.3 are satisfied, that is, if there exists an exhaustive sequence (Kν )ν ⊂ M(G)

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as well as a family {A(ν) : ν ∈ N} of subsets of N satisfying the following conditions: (P1) dens (A(ν)) > 0 for all ν ∈ N. (P2) Each of the families {A(ν); ν ∈ N} and K := {ϕn (Kν ) : n ∈ A(ν), ν ∈ N} is pairwise disjoint. (P3) Given a compact subset K ⊂ G there are only finitely many L ∈ K with K ∩ L 6= ∅. Remark 3.5. Note that condition (P3) in the preceding definition and theorem is stronger than the condition of being frequently runaway given in Proposition 3.1. So frequently exhaustively runaway is a stronger notion as the mere frequently runaway. 4. Vector subspaces of frequently hypercyclic sequences of composition operators As we have mentioned just before, under the condition of being frequently exhaustively runaway we get a high degree of algebraic genericity. This will be shown in the next two theorems. We first consider spaceability. The following technical result will be needed in the proof. Recall that two basic sequences (xn )n , (yn )n in a Banach space (X, sequence (an )n of scalars, the series P∞k · k) are said to be equivalent if, for every P ∞ a x converges if and only if the series n=1 an yn converges. This happens n=1 n n (see [7]) if and only if there exist two constants m, M ∈ (0, +∞) such that J J J

X

X

X





aj xj aj yj ≤ M aj xj ≤ m j=1

j=1

j=1

for all scalars a1 , . . . , aJ and all J ∈ N. By using the first inequality, we are easily led to the next lemma, whose proof can be found in [9, Lemma 2.1]. By L2 (T) we denote the Hilbert space of all Lebesgue-measurable functions f : T → C with  R 1/2 2π 1 dθ iθ finite quadratic norm kf k2 = 2π . It is well known that the |f (e )| 2π 0 sequence (z n )n is a basic sequence in L2 (T).

Lemma 4.1. Assume that G is a domain with D ⊂ G and that (fj )j ⊂ H(G) is 2 j a sequence such that  it is a basic sequence in L (T) that is equivalent to (z )j . If  PJ(l) hl := j=1 cj,l fj is a sequence in span{fj : j ≥ 1} converging in H(G), then l we have J(l) X |cj,l |2 < +∞. sup l∈N j=1

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Theorem 4.2. Let G ⊂ C be a simply connected domain and (ϕn )n be a frequently exhaustively runaway sequence in H1−1 (G). Then the set F HC((Cϕn )n ) is spaceable in H(G). Proof. Let (Kν )ν ⊂ M(G) and {A(ν) : ν ∈ N} be the exhaustive sequence of compact sets and the family of positive integers given by the frequent exhaustive runawayness of (ϕn )n . Without loss of generality, we can assume that D ⊂ K1 ⊂ G, because frequent hypercyclicity is stable under translations and dilations. We are going to modify the proof of Theorem 3.3 appropriately to get spaceability. Let (Pl )l be a dense sequence of H(G) and divide each set A(ν) into infinitely many disjoint subsets A(ν, l, p) (l, p ∈ N), each of them having positive lower density. Observe that condition (P3) of Definition 3.4 implies the existence of k1 ∈ N such that K1 ∩ ϕn (Kν ) = ∅ for all ν ≥ k1 and all n ∈ A(ν). Similarly to the proof of Theorem 3.3, it follows from conditions (P2) and (P3) that the set [ [ F := K1 ∪ ϕn (Kν ) ν≥k1 n∈A(ν)

is a Carleman set of G.

Now, for each µ ∈ N, let  µ  z gµ (z) := Pl (ϕ−1 n (z))  0

us define the function gµ : F → C by if z ∈ K1 if z ∈ ϕn (Kν ), ν ≥ k1 , n ∈ A(ν, l, µ), l ∈ N if z ∈ ϕn (Kν ), ν ≥ k1 , n ∈ A(ν, l, p), p 6= µ, l ∈ N.

It is plain that each gµ is continuous on F and holomorphic in F ◦ , Consequently, by Nersesjan’s Theorem (Theorem 2.2), there exists a function fµ ∈ H(G) such that 1 |fµ (z) − gµ (z)| < µ · min{1, ε(z)} for all z ∈ F, (2) 3 where, again, ε(z) = χ(z, ∂∞ G) as in the proof of Theorem 3.3. We claim that the closed linear space generated by (fµ )µ in H(G), namely, M := span {fµ : µ ∈ N},

is an infinite dimensional closed vector space consisting, except for zero, of frequently hypercyclic functions for (Cϕn )n . With this aim, observe first that, since D ⊂ K1 , we have |fµ (z) − z µ | < 31µ for all z ∈ T. Let (e∗µ )µ be the sequence of coefficient functionals corresponding to the basic sequence (z µ )µ in L2 (T). Since ke∗µ k2 = 1 (µ ∈ N), one obtains ∞ X µ=1

ke∗µ k2

∞ X 1 1 · kfµ − z k2 < = < 1. µ 3 2 µ=1 µ

(3)

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From (3) and the basis perturbation theorem [24, p. 50] it follows that (fµ )µ is also a basic sequence in L2 (T) that is equivalent to (z µ )µ . In particular, the functions fµ (µ ∈ N) are linearly independent and M is an infinite dimensional closed vector space. P Now, fix any f ∈ M \ {0} and let f = µ∈N αµ fµ be its representation in L2 (T). As f 6= 0 there is some nonzero coefficient αµ , which without loss of generality may supposed to be α1 . Furthermore, by the invariance under scalar multiplication of frequent hypercyclicity,we can assume that  α1 = 1. Thanks to P Nj the definition of M, there is a sequence hj := µ=1 αj,µ fµ converging to f in j

H(G). Hence

hj −→ f in L2 (T), because the topology in H(G) is finer than the one in L2 (T). By the continuity of each projection, we obtain that αj,1 → 1 (j → ∞), and without loss of generality we may assume that αj,1 = 1 (if not, just take Hj := hj + (1 − α1,j )f1 ). Finally, by Lemma 4.1, there is a constant H > 0 such that Nj X µ=1

|αj,µ|2 ≤ H for all j ∈ N.

(4)

Next, fix l, ν ∈ N with ν ≥ k1 , n ∈ A(ν, l, 1) and z ∈ ϕn (Kν ). On the one hand, the set ϕn (Kν ) is compact. Then, given δ > 0, there exists j ∈ N such that δ |f (ϕn (z)) − hj (ϕn (z))| < for all z ∈ Kν . (5) 2 But on the other hand, applying (2) and (4) together with the Cauchy–Schwarz inequality guarantees that |hj (ϕn (z)) − Pl (z)| ≤ |f1 (ϕn (z)) − Pl (z)| +

Nj X µ=2

|αj,µ | |fµ(ϕn (z))|

1/2   1/2 Nj Nj X X ≤ ε(ϕn (z)) +  |αj,µ|2   |fµ (ϕn (z))|2  µ=2

< ε(ϕn (z)) + √

√

H

µ=2

∞ X 1 · ε(ϕn (z)) µ 3 µ=2

(6) √ Given ν ∈ N, there is n0 = n0 (ν) ∈ N such that supz∈Kν ε(ϕn (z)) < δ/(2 + 2 H) for n ≥ n0 . Finally, from (5), (6) and the triangle inequality, an argument similar to that of the end of the proof of Theorem 3.3 yields frequent (Cϕn )n -hypercyclicity for f . This concludes the proof.  < (1 +

H) · ε(ϕn (z)).

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A slight modification of the arguments in the proof of the last result allows us to prove the maximal dense lineability of F HC((Cϕn )n ) as well. Theorem 4.3. Let G ⊂ C be a simply connected domain and (ϕn )n be a frequently exhaustively runaway sequence in H1−1 (G). Then the set F HC((Cϕn )n ) is maximal dense-lineable. Proof. Again, let (Kν )ν and {A(ν) : ν ∈ N} be the exhaustive sequence of compact sets and the family of positive integers given by the frequent exhaustive runawayness of (ϕn )n . Without loss of generality we may assume that D ⊂ K1 .

Let (Pl )l be a dense sequence in H(G) and divide each set A(ν) into infinitely many mutually disjoint subsets A(ν, l, p) (l, p ∈ N), each of them with positive lower density. For µ ∈ N fixed, by (P3) there exists kµ ∈ N such that for all ν ≥ kµ and all n ∈ A(ν), we have Kµ ∩ ϕn (Kν ) = ∅. Note that the sequence (kµ )µ ⊂ N can be chosen to be strictly increasing. Similarly to the proofs of Theorems 3.3 and 4.2, one derives from (P2) and (P3) that every set [ [ ϕn (Kν ) Fµ := Kµ ∪ ν≥kµ n∈A(ν)

is a Carleman set. For each µ ∈ N, let us define the function gµ : Fµ+1 → C as follows:   if z ∈ Kµ+1 Pµ (z) −1 gµ (z) := Pl (ϕn (z)) if z ∈ ϕn (Kν ), ν ≥ kµ+1 , n ∈ A(ν, l, µ + 1), l ∈ N  0 if z ∈ ϕn (Kν ), ν ≥ kµ+1 , n ∈ A(ν, l, p), p 6= µ + 1, l ∈ N.

Observe that gµ (z) = 0 for all z ∈ ϕn (Kν ) where n ∈ A(ν, l, 1), ν ≥ kµ+1 and l ∈ N. We are going to use these compact sets later. As in the proof of Theorem 4.2, each gµ is continuous on Fµ and holomorphic on Fµ◦ . Hence, for each µ ∈ N, Theorem 2.2 guarantees the existence of a function fµ ∈ H(G) such that |fµ (z) − gµ (z)|
0, there is some µ0 ∈ N such that |Φ(ϕn (z)) − Pl (z)| < δ/2 for all n ∈ A(ν, l, 1, µ0) and all z ∈ Kν .

Finally, consider the vector subspace of H(G) given by Mmax := Md + Ms .

(8)

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Note that Mmax is dense (because it contains Md ) and has dimension c = dim (H(G)) (because it contains Ms , and dim(Ms ) = c). Fix a function f ∈ Mmax \ {0}. Two cases are possible. If f ∈ Md , then f ∈ F HC((Cϕn )n ). If f 6∈ Md , then there is a function Φ ∈ Ms \ {0} (as in the preceding paragraph) and another function Ψ ∈ Md such that f = Φ + Ψ. Therefore (Cϕn f )(z) = Φ(ϕn (z)) + Ψ(ϕn (z)). Given l, ν ∈ N with ν large enough, we had obtained the existence of α > 0 such that |Ψ(ϕn (z)) − Pl (z)| < α · ε(ϕn (z)) for all z ∈ Kν and all n ∈ A(ν, l, 1) (so for all n ∈ A(ν, l, 1, µ0)). Since supz∈Kν ε(ϕn (z)) → 0 as n → ∞, we get for n ∈ A(ν, l, 1, µ0 ) large enough that |Φ(ϕn (z))| < δ/2 for all z ∈ Kν . Thanks to (8) and the triangle inequality, we get |(Cϕn f )(z) −Pl (z)| < δ for the same n, z. Then the proof of the fact f ∈ F HC((Cϕn )n ) is concluded as soon as we take into account the density of (Pl )l in H(G) and the property dens (A(ν, l, 1, µ0 )) > 0.  Remark 4.4. Let G ⊂ C be any domain of C and (ϕn )n ⊂ H(G, G), and let us consider the family F HCM ((Cϕn )n ) := {f ∈ H(G) : dens({n ∈ N : f ◦ ϕn ∈ BK (g, δ)}) > 0 for each K ∈ M(G), each δ > 0 and each g ∈ H(G)}. Under the same assumptions as in Theorems 3.3-4.2-4.3 (the frequent exhaustive runawayness of (ϕn )n ), and with the same proofs, it is obtained that the set F HCM ((Cϕn )n ) is spaceable and maximal dense lineable in H(G). Note that, if G is multiply connected, then F HCM ((Cϕn )n ) is bigger than F HC((Cϕn )n ). In fact, if G is a finitely connected domain that is not simply connected and (ϕn )n ⊂ H1−1 (G), then the set HC((Cϕn )n ) (so the set F HC((Cϕn )n ) as well) is empty: see [28, Theorem 3.15]. 5. Examples 1. Consider the slit complex plane G := C \ (−∞, 0]. Let α, β ∈ R with β > 0 and β ≥ 1 + α. For N ∈ N, z 1/N will represent the principal branch of the Nth root of z in G. Fix N ∈ N. Then the mappings ϕn (z) := nα z 1/N + nβ (n ∈ N)

belong to H1−1 (G) \ Aut (G). Let C > 0 be a constant to be specified later, and consider the numbers Rν = (C ν β−α )N and the compact sets n o π iθ Kν = z = re : min{Rν , 1/N} ≤ r ≤ Rν , |θ| ≤ (1 − (1/ν)) (ν ∈ N). N Note that Kν ⊂ D(0, Rν ), the closed disc with center 0 and radius Rν . Hence 1/N ϕn (Kν ) ⊂ D(nβ , nα Rν ). Plainly, all Kν ’s lack holes and form an exhaustive sequence of compact subsets of C \ (−∞, 0]. By [19, Lemma 2.2], there exist pairwise disjoint subsets A(l, ν) (l, ν ≥ 1) of N with dens(A(l, ν)) > 0 such that, for any n ∈ A(l, ν) andSm ∈ A(k, µ), we have n ≥ ν and |n − m| ≥ ν + µ if n 6= m. Define A(ν) := l∈N A(l, ν). Of course, dens(A(ν)) > 0 for every ν ∈ N, and the family {A(ν) : ν ∈ N} is pairwise disjoint. Pick two distinct members A, B ∈ K := {ϕn (Kν ) : n ∈ A(ν), ν ∈ N}. Then there are distinct m, n ∈ N (m >

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n, say) as well as sets A(l, ν), A(k, µ) such that n ∈ A(l, ν), m ∈ A(k, µ), A = tβ −1 ϕm (Kµ ) and B = ϕn (Kν ). It is easy to see that σ := inf t>1 tα (t−1) β−α > 0. β

−1 Take C := min{1/2, σ/4}. Since m > n, we get (m/n)(m/n) α ((m/n)−1)β−α ≥ 4 C, that is, mβ − nβ ≥ 4 C · mα (m − n)β−α . The distance between the centers of the discs 1/N 1/N D1 := D(mβ , mα Rµ ) and D2 := D(nβ , nα Rν ) is mβ − nβ . But observe that

mα + nα mα + nα · (m − n)β−α ≥ 4 C · · (µ + ν)β−α 2 2 mα + nα ≥ 2C · · (µβ−α + ν β−α ) ≥ mα C µβ−α + nα C ν β−α 2 = radius (D1 ) + radius (D2 ).

mβ − nβ > 4 C ·

Consequently, D1 ∩ D2 = ∅ and so A ∩ B = ∅. Then K is pairwise disjoint. Finally, fix a compact set K ⊂ C \ (−∞, 0]. There is R > 0 such that K ⊂ D(0, R). If L ∈ K, there exists (l, ν) ∈ N2 and n ∈ A(l, ν) (so n ≥ ν) with L = 1/N 1/N ϕn (Kν ) ⊂ D(nβ , nα Rν ). It follows that each w ∈ L satisfies |w − nβ | ≤ nα Rν , 1/N hence (recall that C ≤ 1/2) we get |w| ≥ nβ − nα Rν = nβ − nα C ν β−α ≥ β α β−α β n −n Cn ≥ (1/2)n > R if n is large enough. Therefore K ∩ ϕn (Kν ) = ∅, so K ∩ L = ∅ except for finitely many L ∈ K, because no n may belong to two different A(l, ν)’s. According with Theorems 3.3-4.2-4.3, (Cϕn )n is frequently hypercyclic and the set F HC((Cϕn )n ) is spaceable and maximal dense-lineable in H(C \ (−∞, 0]).

2. If G and Ω are simply connected domains different from C, then the Riemann conformal representation theorem (see, e.g., [1]) provides an isomorphism (that is, a bijective biholomorphic mapping) f : G → Ω. Therefore, if ϕ ∈ H(G, G), we have f ◦ ϕ ◦ f −1 ∈ H(Ω, Ω) and the mapping h ∈ H(Ω) 7→ h ◦ f ∈ H(G) is a linear homeomorphism. Hence every example of a sequence (ϕn )n ⊂ H(G, G) satisfying that F HC((Cϕn )n ) is nonempty (spaceable, maximal dense-lineable, resp.) provides us with an example of a sequence (Φn )n ⊂ H(Ω) such that F HC((CΦn )n ) is nonempty (spaceable, maximal dense-lineable, resp.). Indeed, take Φn := f ◦ ϕn ◦ f −1 (n ∈ N). For instance, we have in accordance with Example 1 that F HC((Cϕn )n ) is spaceable and maximal dense-lineable in H(C \ (−∞, 0]), where ϕn (z) = n + z 1/N . Moreover, the mapping f : z ∈
2 1/2 C \ (−∞, 0] 7→ zz 1/2 −i ∈ D is an isomorphism with inverse f −1 (z) = − z+1 . z−1 +i Consequently, the set F HC((CΦn )n ) is spaceable and maximal dense-lineable in
1/2 z+1 2 1/N ) ) −i n+(−( z−1 H(D), where Φn (z) =
1/2 . z+1 2 1/N n+(−( z−1 ) )

+i

3. Let a > 0 and γ ≥ 1. It is easy to check that every linear fractional function 2(z−1) Φn (z) := 1 + 2−an γ (z−1) (n ∈ N) is an automorphism of D. Moreover, every Φn is parabolic (see [38, pp. 8–9]), that is, it has a unique fixed point, located at T

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(namely, at 1). With arguments similar – but easier – to those given in Example 1, we can check that if γ ≥ 1, Π+ := {z ∈ C : Re z > 0} is the open right half-plane and ϕn (z) := z + i anγ for each n ∈ N (note that (ϕn )n ⊂ Aut (Π+ )), then F HC((Cϕn )n ) is spaceable and maximal dense-lineable in H(Π+ ). Since 1+z z−1 f (z) := 1−z is an isomorphism between D and Π+ with inverse z+1 , and since −1 Φn = f ◦ϕn ◦f , we obtain as in Example 2 the spaceability as well as the maximal dense-lineability for F HC((CΦn )n ) in H(D). The case γ = 1 gives the same properties (spaceability was already known, see Section 2) for the set F HC(Cϕ ), 2(z−1) . where ϕ is the parabolic automorphism of D given by ϕ(z) = 1 + 2−a(z−1) 4. Let G = C. In this case H1−1 (G) = Aut (C) = {similarities z 7→ az + b : a, b ∈ C, a 6= 0}. Examples of sequences of similarities (ϕn (z) = an z + bn )n for which (Cϕn )n is frequently hypercyclic were provided in [10, Theorem 4.1] (see Section 1), whose proof inspired in fact the one of our Theorem 3.3. Since the assumptions of all three Theorems 3.3-4.2-4.3 are the same, we obtain that for such sequences (ϕn )n the sets F HC((Cϕn )n ) are even spaceable and maximal dense-lineable in H(C). 5. In this final example we do not use Theorems 3.3-4.2-4.3. Assume that G ⊂ C is a Jordan domain, that is, a domain for which ∂∞ G is a homeomorphic image of T (hence G need not be bounded). As a particular instance of a result by B`es [17] mentioned in Section 2, the set F HC(Cϕ ) is spaceable if it is assumed that ϕ ∈ H1−1 (G) and there is ξ ∈ ∂∞ G such that ϕ[n] → ξ (n → ∞) uniformly on compacta in G (indeed, such a ϕ cannot have fixed points in G). According to Osgood–Carath´eodory’s theorem (see [30]), there is an isomorphism f : G → D ∞ ∞ that is extendable to a homeomorphism G → D, where G is the closure in C∞ of G. Since the polynomials form a dense set in H(D), the set D := {P ◦ f : P polynomial} is a dense vector subspace of H(G). Now, the sequence ((Cϕ )[n] Q)n converges in H(G) for every Q ∈ D, because if Q = P ◦ f then (Cϕ )[n] Q = Cϕ[n] Q = P ◦ f ◦ ϕ[n] −→ P (f (ξ)) (n → ∞) uniformly on compacta in G. Since F HC(Cϕ ) = F HC((Cϕ[n] )n ) is spaceable and H(G) is a separable F-space, from [14, Theorem 4.10(a)] we can conclude that F HC(Cϕ ) is maximal dense-lineable in H(G). Hence, in particular, the set F HC(Cϕ ) is maximal dense-lineable not only for the parabolic automorphism ϕ as defined in the last line of Example 3, but also for each parabolic automorphism of D (since each parabolic automorphism of D exactly has one fixed point which lies on T and to which its iterates converge locally uniformly on D). Acknowledgements. The first, second and fourth authors have been partially supported by Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127

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Grant P08-FQM-03543 and by MEC Grant MTM2015-65242-C2-1-P. The third author has been supported by DFG-Forschungsstipendium JU 3067/1-1.

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´lisis Matema ´tico Departamento de Ana ´ticas, Facultad de Matema Universidad de Sevilla Avda. Reina Mercedes s/n 41080 Sevilla, Spain. E-mail address: [email protected] ´lisis Matema ´tico Departamento de Ana ´ticas, Facultad de Matema Universidad de Sevilla Avda. Reina Mercedes s/n 41080 Sevilla, Spain. E-mail address: [email protected] Fachbereich IV Mathematik ¨t Trier Universita D-54286 Trier, Germany E-mail address: [email protected] ´lisis Matema ´tico Departamento de Ana ´ticas, Facultad de Matema Universidad de Sevilla Avda. Reina Mercedes s/n 41080 Sevilla, Spain. E-mail address: [email protected]