1. Introduction

This paper is the fourth one in a series of papers, [24], [34], [35], which aims to a better understanding of the phenomenon of common hypercyclic vectors for uncountable many families of hypercyclic operators of translation type. The notion of hypercyclicity has been studied intensively the last 30 years and there is by now a well developed theory on this subject, see for instance the two important recent books [9], [27] and the survey of Gross-Erdmann [26]. Let us recall the relevant definitions. A sequence (T_n) of continuous and linear operators acting on a real or complex topological vector space X is called hypercyclic, if there exists a vector x∈X so that the set {T_nx : n = 1, 2, . . .} is dense in X; in this case x is called hypercyclic for (T_n) and the symbol H C (T_n) stands for the set of all hypercyclic vectors for (T_n). Let T: X → X be a linear and continuous operator.

We define the iterates of T inductively as follows: T¹:= T and T ⁿ⁺¹= Tⁿo T , n = 1, 2, . . . where with Tⁿo T we mean the usual composition of the operators T ⁿ and T for n = 1, 2, . . . . If in the previous definition, the sequence (T_n) comes from the iterates of a single operator T, i.e. T_n = T ⁿ, n = 1, 2, . . . then T is called hypercyclic, x is called hypercyclic for T and H C (T) denotes the set of hypercyclic vectors for T. Observe that, in the above situation the topological vector space X is necessarily separable. For several examples of hypercyclic operators including many classical operators, such as weighted shifts, differential operators, adjoints of multipliers and so on, we refer to [9], [27]. We denote C, the set of complex numbers and H(C) the set of entire functions that is supposed endowed with the topology T_u of local uniform convergence from now on.

Our interest here lies on a particular operator, the translation operator acting on the space H(C). Let a ∈ C. For obvious reasons, the operator T_a: H(C) → H(C) defined by T_a (f)(z):= f (z + a), for f ∈ H(C), z ∈ C, is called translation. A classical result due to Birkhoff [13] says that T₁ is hypercyclic. This means, that there exists an entire function f whose positive integer translates approximate every entire function, i.e. the set {f (z + n): n = 1, 2, . . .} is dense in (H(C), T_u). Actually for every a∈C∖{0} the translation operator T_α is hypercyclic. As an easy application of Baire’s category theorem we have the following dichotomy: if X is a separable topological vector space and T is a linear and continuous operator on X then either H C (T ) = ∅ or H C (T) is G_δ and dense set in X, see [9], [27]. Recall that a subset A of X is called G_δ if it can be written as countable intersection of open sets. Therefore, for every α ∈ C \{0} the set H C (T_α) is G_δ and dense set in (H (C)T_u) and as an immediate consequence of Baire’s category theorem, we have that the set is non-empty for every sequence of non-zero complex numbers. Our point of departure is the following extension of Birkhoff ’s theorem due to Costakis and Sambarino [23]:

The difficulty of proving such a result is, of course, the uncountable range of a. Subsequently Costakis, in an attempt to generalize the previous result established the following:

Theorem 1.1. [20] Let (λ_n) be a sequence of non-zero complex numbers with

|λ_n | → +∞ which also satisfies the following condition (Σ):

Condition (Σ). For every M > 0 there exists a subsequence (µ_n) of (λ_n) such that

The purpose of the present work is to extend the above theorem by allowing a strictly wider class of sequences (λ_n). However, it is necessary to impose certain restrictions on (λ_n) so that the conclusion of the above theorem holds, as the following result from [24] shows: if (λ_n) is a sequence of non-zero complex numbers with then

Frederic Bayart improved this result by proving that

We consider a sequence (λ_n) of complex numbers that satisfies the following condition which we call it condition (Σ₁) from now on.

Condition (Σ₁). For every M > 0 there exists a subsequence (µ_n) of (λ_n) such that

(i) µ₁ ≠ 0

(ii) |µ_n+1 | − |µ_n| > M for every n = 1, 2, . . . and

(iii) We denote A₂ the sequences that satisfy the previous condition (Σ₁). At this point it is instructive to observe that the sequences (n²), (n³), . . ., satisfy condition (Σ₁) but not condition (Σ) of Theorem 1.1.

We denote by the circle with center 0 and radius r, for some r > 0.

Theorem 1.2. Fix a sequence of complex numbers Λ=(λ_n) that satisfies the above condition (Σ₁). Then for every r ∈ (0, +∞) the set

is a G_δ and dense set in (H(C), T_u). In particular,

It is clear that property (iii) of condition (Σ₁) relaxes the corresponding condition (ii) of (Σ).

This differentiate Condition (Σ₁) from condition (Σ) because for a sequence |µ_n | from non-zero complex numbers the series

has always a unique limit, finite or + infinity. This does not hold for a sequence of the form

as someone can see, using elementary calculus because such a sequence can have many limit points.

We note that Frederic Bayart examined in [2] similar problems in Rⁿ generally in his significant paper.

One may also wonder if property (iii) of (Σ₁) in Theorem 1.2 can be relaxed by replacing the lim with lim sup. We prove now that this has no sense.

More specifically:

We denote N, the set of natural numbers.

We consider the set A₃ of sequences (λ_n) of complex numbers that satisfy the following condition which we call it Condition (Σ₂) from now on.

Condition (Σ₂). For every M>0 there exists a subsequence (µ_n) of (λ_n) such that:

(i) µ₁ ≠ 0

(ii) |µ_n+1 | − |µ_n| > M for every n = 1, 2, . . . and

(iii)

Lemma 1.3. It holds: A₂ = A₃.

Proof. It is obvious that

(1.1)

We prove now the reverse inclusion.

Let (λ_n) ∈ A₃. We fix a positive number M. Then, there exists a subsequence (µ_n) of (λ_n) that satisfies the three properties (i), (ii) and (iii) of condition (Σ₂).

We distinguish two cases.

If

then the sequence (µ_n) satisfies of course property (iii) of condition (Σ₁).

Of course a_n →0, as n→ + ∞.

By property (iii) of (Σ₂) there exists n₁ ∊ N such that:

(1.2)

Because there exists

(1.3)

By (1.2) and (1.3) we get:

(1.4)

By property (iii) of (Σ₂) there exists n₃∈N, n₃ > n₂ such that:

(1.5)

Because there exists n₄ ∊ N, n₄ > n₃ such that

(1.6)

By (1.5) and (1.6) we get:

(1.7)

Inductively we construct a subsequence such that:

Of course the subsequence satisfies the properties (i), (ii) and (iii) of (Σ₁). This shows that A₃ ⊂ A₂, that completes the proof.

It is important to mention that in [35] we obtain a full strength of the conclusion of Theorem 1.1, namely we show that under the assumptions of Theorem 1.1 the set

is G_δ and dense set in (H(C), T_u). On the other hand, relaxing condition (iii) of Theorem 1.1 as above, the price we pay, at least for now, is the “thin”, but still uncountable, range of a in the conclusion of Theorem 1.2. A further connection of the present work with our main result from [34] will be discussed in Section 5.

There are several recent results concerning either the existence or the non-existence of common hypercyclic vectors for uncountable families of operators, see for instance, [1]-[12], [14]-[25], [27]-[29], [31]-[35].

The paper is organized as follows. Sections 2-4 contain the proof of Theorem 1.2. In the last section, Section 5, we connect our work with the main results from [34], [35].

2. Three Basic Lemmas

We fix a positive number r₀.

Let us now describe the main steps for the proof of Theorem 1.2. Defining the arcs

and using Baire’s category theorem we easily see that Theorem 1.2 reduces to the following

Proposition 2.1. Fix a sequence (λ_n) of complex numbers that satisfies the above condition (Σ₁). Fix three real numbers r₀, θ₀, θ_T such that r₀ ∈ (0,+∞),

For the proof of Proposition 2.1 we introduce some notation which will be carried out throughout this paper. Let (p_j), j = 1, 2, . . . be a dense sequence of (H(C), T_u), (for instance, all the polynomials in one complex variable with coefficients in (Q + iQ) where Q is the set of rational number. For every m, j, s, k ∈ N we consider the set

Clearly, Baire’s category theorem and the three lemmas stated below imply Proposition 2.1.

Lemma 2.1

Lemma 2.2. For every m,j,s,k ∊ N the set E (m,j,s,k) is open set in (H(C), T_u).

Lemma 2.3.

The proof of Lemma 2.1 is in [34]. The proof of Lemma 2.2 is similar to that of Lemma 9 in [22] and it is omitted. It remains to prove Lemma 2.3. This will be done in Sections 3 and 4 [23].

3. Construction of the Partition and the Disks

Our main task is to prove Lemma 2.3. This means of course to find a natural number m and an entire function f such that the inequality that is described in the set E(m, j, s, k) is satisfied for every α ∊ A. If the set A is finite this can be done easily if we take different terms from the sequence (λ_n) that are far away each-other. However, if A is uncountable as an arc we cannot apply the previous method.

We remark that if the inequality is satisfied for some value α ∈Α, then the same inequality is satisfied for values of a that are close to a. So the basic idea is to choose a suitable partition from finite many values of a that satisfy the desired inequality and wait that all the others satisfy the some inequality with the same term of the sequence (λ_n) of some of the finite many prescribed values.

In a point of the proof it is needed, for technical reasons to have that the

example for the sequence λ_n = n², , n = 1, 2, ….

For this reason the method in [20] and [23] cannot be applied here. However, if it is permitted to to become large enough. This exactly is the key point of our proof here.

After we choose suitable terms of the sequence (λ_n) and some partition from values a∈A and we apply Runge’s Theorem on convenient discs with center in the point λ_na and the same radius R that is assigned by the problem.

More specifically:

Let Λ = (λ_n) be a sequence of non-zero complex numbers such that satisfies condition (Σ₁). Let three real numbers θ₀, θ_T , r₀ be as in Proposition 2.1. For the sequel we fix four positive numbers c₁, c₂, c₃, c₄ such that c₂ ∊ (0, 1), c₃ > 1, c₄ > 1, where . After the definition of the above numbers we fix a subsequence (μ_n) of (λ_n) such that:

Throughout Section 3 the positive integer m₀ will appear frequently and it is fixed from now on.

3.1. Step 1. Partitions of the Interval [θ₀, θ_T]

For every sufficiently large positive integer m (actually m ≥ m₀) we shall construct a corresponding partition ∆_m of [θ₀, θ_T]. For every m ≥ m₀ let m₁(m) be the minimum positive integer such that

(3.1)

Obviously m₁(m) ≥ m+1 for every m=m₀, m₀ + 1. . ., since c₃ > 1.

Let m be any positive integer with .

We define the numbers or in a more compact form

(3.2)

We denote

Now let some positive integer Then there exists a unique pair , where such that:

ν = k(m₁(m) − m + 1) + j.

Define

with respect to ν. So there exists the maximum natural number ν_m ∈ N such that We set

3.2. Step 2. A Lower Bound for the Stopping Time ν_m

Lemma 3.1.

In particular ν_m ≥ m₁(m) − m + 1.

Proof. By the definition of the numbers θ_j^(m) j = 0, 1, . . ., m₁(m) − m+ 1 we have

(3.3)

In order to bound the right hand term in the equality above, observe that

(3.4)

which follows from the definition of the number m₁(m). Since c₁ = 4(c₃+ 1)

c₂ ∈ (0, 1) and |µ_m | > c₁, we deduce that

(3.5)

3.3. Step 3. Partitions of the Arc

Consider the function given by

and for r₀ > 0 we define the corresponding curve by

For any given positive integer defines a partition of the , where m₀ is the positive integer defined above in Step 1 and ∆_m is the partition of the interval [θ₀, θ_T] constructed in Step 1. For every m ∈ N with m ≥ m₀ define

which we call partition of the

3.4. Step 4. Construction of the Disks

Our task in this subsection is to assign to each point w of the partition Pm for m≥ m₀ a suitable closed disk with center wµ(w) and radius c₄ (the radius will be the same for every member of the family of the disks), where µ(w) will be chosen from the sequence (µ_n). We shall see that, the construction of the partition P_m ensures on the one hand that the points of the partition are close enough to each other on the arc A and on the other hand that the disks centered at the points wµ(w) for w ∈P_m with fixed radius c₄ are pairwise disjoint.

We set

Β:= {z ϵ C/|z| ≤ c₄}

and fix any positive integer m with m ≥ m₀. Let w be an arbitrary point in P_m. There exists unique n ∈ {0, 1, . . ., ν_m} such that . Now there exists unique k ∈N ∪ {0}, and j ∈ {0, 1, . . ., m₁ (m) − m} such that n = k (m₁ (m) − m + 1) + j and define

µ (w) := µ_m+j.

Thus we assign, in a unique way, a term of the sequence (µ_n) to every point of P_m and more specifically a term of the finite set and we introduce the notation

B_w:= B + wµ(w),

to take our construction. The desired disks are the disks B and B_w, w ∈ P_m. Denote by

D_m := {B} ∪ {B_w : w ∈ P_m }

the collection of the above disks.

Remark. Since µ(w) = µ_m+j does not depend on K, this means that several centers wµ(w) are on the same disk centered in 0.

3.5. Step 5. The Disks are Pairwise Disjoint

Lemma 3.2. Let with m≥ m₀. Then for every

Proof. account c₁ = 4 (c₃+1) > 2c₃ we get

(3.6)

Let w∈Pm . The closed disks B, B_w are centered at 0, wµ (w) respectively and they have the same radius c₄. Hence, we have to show that |wµ (w)| > 2c₄

Observe now that, by the definition of µ (w) in the previous subsection,

(3.7)

for some positive integer n∈N. As a consequence of the definition of the sequence (µ_n) we have

(3.8)

the proof of the lemma.

Lemma 3.3. Let m ∈ N with m ≥ m₀ and w₁, w₂ ∈ P_m with w₁ ≠ w₂. Then

Proof. We distinguish two cases:

(i) |µ (w₁)| < |µ (w₂)|.

Our hypothesis implies

|w2µ(w2)−w1 µ(w1)| ≥ ||w2 µ(w2)|−|w1µ(w1 )|| = r0(|µ(w2)|−|µ(w1 )|) ≥ r0 c1 > 2c4 ,

where the last inequality above follows by (3.6).

(ii) |µ (w₁)| = |µ (w₂)|.

for some n₁ , n₂ ∈{0, 1, . . ., ν_m} and n₁ ≠ n₂ because w₁ ≠ w₂. Without loss of generality we suppose that n₁ < n₂.

Now there exists a unique pair (k₁, j₁), where k₁ ∈ N ∪ { 0}, j₁∈ { 0, 1, . . ., m₁(m) − m} and a unique pair (k₂, j₂) where k₂ ∈ N ∪ {0} and j₂ ∈ {0, 1, . . ., m₁(m) − m} such that

(3.9)

and

(3.10)

By definition we have and the hypothesis yields

|µ(w1)| = |µ(w2)| ⇔ µ(w1) = µ(w2).

So we have j₁ = j₂. Thus

and

(3.11)

By (3.9), (3.10) and the fact that n₁ <n₂ and j₁= j₂ we have k₁ < k₂ ⇒ k₂ ≥ k₁ + 1.

Using now (3.11) it follows that

(3.12)

A lower bound for the quantity |w₂μ (w₂) − w₁μ (w₁)| is:

(3.13)

The last inequality implies the last equality of (3.13).

Consider Jordan’s inequality

We have

(3.14)

Now, inequalities (3.12), (3.13) and (3.14) imply

(3.15)

By the definition of the number σm and relation (3.3) of Lemma 3.1 we obtain

The last equality, inequality (3.15) and the definition of the number m₁(m) give

and the properties of our fixed numbers imply 4r₀c₂c₃ > 2c₄.

It follows that B_w1 ∩ B_w2 = ∅ and the proof of this lemma is complete.

By Lemmas 3.2, 3.3 we conclude the following.

Corollary 3.1. For every positive integer m with m ≥ m₀ the family D_m := {B}∪{B_w : w ∈P_m} consists of pairwise disjoint disks.

4. Proof of Lemma 2.3

dense in (H(C), T_u). For simplicity we write . Consider fixed g ∈ H(C), a compact set K ⊆ C and . We seek f ∈ H(C) and a positive integer such that

(4.1)

and

(4.2)

Fix R₁ > 1 sufficiently large so that

K ∪ {z ∈ C| |z| ≤ k₁} ⊂ {z ∈ C| |z| ≤ R₁}.

and then choose 0 < δ₀ < 2 such that

(4.3)

We set

Observe that c₃, c₄ ∈ (1, +∞) and c₂ ∈ (0, 1).

After the definition of the above numbers we choose a subsequence (µ_n) of (λ_n) such that

|µ_n |, |µ_n+1 | − |µ_n | > c₁ for n = 1, 2, . . .

and

By condition (ii), there exists a positive integer m₀ such that for every m ≥ m₀

After that, on the basis of the fixed numbers r₀, θ₀, θ_T , c₁, c₂ , c₃, c₄ and m₀, for every positive integer m with m ≥ m₀ we define the corresponding partition P_m (see section 3). From now on till the end of the proof of the lemma we fix a positive integer m ≥ m₀ with its corresponding partition P_m. For simplicity we write

P := P_m .

Then, we define the set L as follows:

where the discs B_w , w ∈ P, are constructed in Section 3. By Corollary 3.1, the family D : = {B } ∪ {B_w: w ∈ P } consists of pairwise disjoint disks. Therefore the compact set L has connected complement. This property is needed in order to apply Mergelyan’s theorem [30]. We now define the function h on the compact set L, by

By Mergelyan’s theorem [29] there exists an entire function f (in fact a polynomial) such that

(4.4)

By the definition of h, (4.4) and the definitions of sets Κ and B, where Κ ⊆ B, it follows that

which implies the desired inequality (4.2).

It remains to show (4.1). Let α ∈ A. There exists a unique θ ∈ [θ₀, θ_T ] such that α = r₀ e^2πiθ . Now there exists unique ρ ∈ {0, 1, . . ., ν_m − 1} such that:

and we then define

For the above, recall that the definitions of ν_m and , are defined in Section 3. Set w₀: = r₀e2πiθ₁ ∈ P. We shall prove that for every z ∈ C, |z | ≤ R₁ we have z + αµ (w₀ ) ∈ B_w0 .

(4.5)

For |z | ≤ R₁ we have:

(4.6)

By (4.5) and (4.6) it suffices to prove

(4.7)

We have:

which implies (4.7). So, for every z ∈ C, |z | ≤ R₁

(4.8)

The definition of h and (4.8) give that for every z ∈ C, |z | ≤ R1

(4.9)

By (4.3) and (4.7) we get: for every z ∈ C, |z| ≤ R₁

(4.10)

The triangle inequality and the fact that k₁ ≤ R₁ give

(4.11)

Setting

m₁: = max {n ∈ N|λ_n = µ (w), for some w ∈ P },

observing that the definition of m₁ is independent of α ∈ A and in view of (4.11) we conclude that for every α ∈ A there exists some n ∈ N with n ≤ m₁ such that

where f ∈ H (C), since f is a polynomial. This implies (4.1) and the proof of the lemma is complete.

5. Examples of Sequences Λ: = (λ_n) Satisfying Condition (Σ₁)

In this section we show that our main theorem is not covered by our recent results in [34], [35]. Recall that a sequence of non-zero complex numbers (λ_n) with |λ_n| → ∞ satisfies condition (Σ) if:

for every M > 0 there exists a subsequence (µ_n ) of (λ_n) such that

For instance, sequences of linear growth i.e. λ_n = αn + b, α, b ∈ C, α ≠ 0, n = 1, 2, . . ., satisfy condition (Σ), or sequences (λ_n), such that

We introduce the following definitions.

For Λ ∈ L’ , define

and

i(Λ):= inf B (Λ)

Clearly,

i (Λ) ∈ [1, +∞] for every Λ ∈ L′.

Our main results in [34], [35] are the following

a G_δ and dense subset of (H(C), T_u).

dense subset of (H(C), T_u).

In order to come closer to a complete picture of our investigations we introduce one last class of sequences. Set

A₄ := {Λ = (λ_n) ∈ L′|i (Λ) = 1}.

Proposition 5.1. A₄ ⊂ A₂

Proof. Let Λ = (λ_n ) ∈ A₄ and fix a positive number M. By Lemma 7.1 in [34] there exists a subsequence (µ_n) of (λ_n) such that |µ_n+1| − |µ_n | > M for every n = 1, 2, . . . and |µ_n+1|/|µ_n |→1 as n→ + ∞. . To this end, let A >0 and consider any positive number ε so that ε < 1/A. Since |µ_n+1 |/|µ_n|→1 as n→ + ∞ there exists a positive integer N such that |µ_n /µ_n+1 | > 1/(1 + ε) for every n ≥N.

Repeated using of the previous inequality gives

and upon summation we get

and this completes the proof.

From the above we have A₁ ∪ A₄ ⊂ A₂.

In view of Theorem 5.1 and in order to completely characterize the sequences such that the set (H(C), T_u) one has to deal with sequences Λ ∈ L for which i (Λ) > 1. This is one of the reasons we introduced the classes A₁, A₂. Indeed, it is established in [35] that the class A₁ contains sequences Λ ∈ L with i (Λ) > 1. On the other hand, there exist sequences Λ ∈ L′ with i (Λ) = 1 and Λ ∉ A₁, see [35]. Since,

A₁ ⊂ A₂

we conclude that the class A₂ contains sequences Λ ∈ L′ with i (Λ) > 1. The above inclusion is strict; for instance, the sequence λ_n = n², n = 1, 2, . . ., belongs to A₂ but not in A₁ . However i ((n² )) = 1, therefore for this sequence the conclusion of Theorem 5.1 holds and of course in this case the conclusion of Theorem 1.2 is covered by the much stronger Theorem 5.1. So the interest here is to show that there exists Λ ∈ A₂\A₁ with i (Λ) = M for some positive real number M > 1, and this in turn shows that our main result, Theorem 1.2, is not covered by Theorems 5.1, 5.2. This is the content of the following

Proposition 5.2. Fix some M > 1. There exists Λ ∈ A₂\A₁ such that i (Λ) = M.

Proof. We construct inductively a countable family {F_n }, n = 1, 2, . . . of finite sets F_n ⊂ [1, +∞) according to the following rules. Firstly, we fix the sequence where with [α_n ] we mean the integer part of the real number α_n .

(i) F₁ = {1}.

(ii) F_m = {(α_m + ν )²| ν = 0, 1, . . ., [α_m] + 1} m = 1, 2, . . ..

(iii) min F_m+1 = M · max F_m for each m = 1, 2, . . .,

F_m1 ∩ F_m2 = 0. Set

We define the sequence Λ = (λ_n) to be the enumeration of by its natural order.

strictly increasing sequence of positive numbers. We divide the proof into several claims.

Claim 1. For every subsequence (µn) of Λ we have

Proof. Let (µ_n) be a fixed subsequence of Λ. We prove that for every natural number n₁ ∈ N, there exists N ∈ N with N ≥ n₁ such that

So, fix n₁ ∈ N. Let m₂ be the unique positive integer such that . We set . It is obvious that . We set . Then and so .

So we proved that for every n ∈ N, there exists some N ≥ n such that

then by the construction of we remark by definition of that for some ; thus

(5.1)

(5.2)

By (5.1),(5.2) and since the conclusion follows. This completes the proof of Claim 2.

Claims 1 and 2 imply that i(Λ)=M. We now show the following claim.

and by noticing that

The above claim shows that . We now show our last claim

Claim 4.

Proof. Since previously, we have trivially

Finally, since we can deduce that the sequence satisfies condition (C). Hence and the proof of this proposition is complete.

Proposition 5.2 shows that the previous inclusion A₁ ∪ A₄ ⊂ A₂ is strict. So, at the level of a circle the main result of this paper is “strictly” stronger than the union of the main results in [34], [35].

ACKNOWLEDGEMENTS

I am grateful to George Costakis for his helpful comments and remarks and for all the help he offered me concerning the presentation of this work.