Introduction

A continuous linear operator T defined on a Fréchet space \(\mathcal {X}\) is said to be hypercyclic provided there exists a vector \(f\in \mathcal {X}\) such that \(\{T^nf\}\) is dense in \(\mathcal {X}\) (we refer to the monographs [4, 14]). Let us denote by \(\mathcal {H}({\mathbb {C}})\) the space of entire functions endowed with the topology of uniform convergence on compact subsets. In 1929, G. D. Birkhoff [9] proved that the translation operator is hypercyclic on \(\mathcal {H}({\mathbb {C}})\) and some years later G. R. MacLane proved that the differentiation operator is hypercyclic on \(\mathcal {H}({\mathbb {C}})\). These seem to be the first results on hypercyclic operators.

Along the paper, \(\phi (z)=\sum _{n=0}^\infty \beta _nz^n\) will be a nonzero entire function of exponential type: there are constants \(A, B>0\) such that \(|\phi (z)|\le Ae^{B|z|}\) for all \(z\in \mathbb {C}\). We will denote by \(\text {span}\, (A)\) the subspace generated by a subset A of a vector space. In 1991, G. Godefroy and J. H. Shapiro ([13]) improved Birkhoff and MacLane’s results by proving that if T commutes with the differentiation operator D, then \(T=\phi (D)\) where \(\phi\) is an entire function of exponential type and \(T=\phi (D)\) is hypercyclic on \(\mathcal {H}({\mathbb {C}})\) if and only if T is not a multiple of the identity.

Throughout the paper, all operators will be considered linear and continuous on the space \(\mathcal {H}({\mathbb {C}})\) endowed with the compact open topology. A complex number \(\lambda\) is called an extended eigenvalue of an operator T if there exists a nonzero operator X, which is called an extended \(\lambda\)-eigenoperator of T, such that \(TX=\lambda XT\). Assume that T is an extended \(\lambda\)-eigenoperator of the differentiation operator D. In [5], it was characterized when a extended \(\lambda\)-eigenoperator of the differentiation operator D is hypercyclic, extending results by Godefroy and Shapiro ([13]), Aron and Markose ([1]), and Bernal and Montes ([7]). This is the result in [5].

Theorem 1.1

(See [5]). An operator \(T\,:\mathcal {H}({\mathbb {C}})\rightarrow \mathcal {H}({\mathbb {C}})\) is an extended \(\lambda\)-eigenoperator of D if and only if T factors as \(T=R_\lambda \phi (D)\), where \(R_{\lambda }f(z)=f(\lambda z)\) is the dilation operator and \(\phi\) is an entire function of exponential type. Moreover, the following statements are equivalent for \(T=R_\lambda \phi (D)\) when \(\lambda \ne 1\):

  1. (a)

    The operator T is hypercyclic.

  2. (b)

    The operator T is topologically mixing.

  3. (c)

    The zero set of \(\phi\) is nontrivial (i.e., \(\phi ^{-1}(0)\ne \emptyset , \mathbb {C}\)) and \(|\lambda |\ge 1\).

Let us recall that an operator T is called topologically mixing if for any nonempty open subsets U, V, there exists \(N\in \mathbb {N}\) such that \(T^n(U)\cap V\ne \emptyset\) for all \(n\ge N\) (equivalently for any subsequence of natural numbers \((n_k) \subset \mathbb {N}\) there exists a function f such that \(\{T^{n_k} f\}_{k\ge 1}\) is dense in \(\mathcal {H}({\mathbb {C}})\)).

An operator T is frequently hypercyclic if there exists a vector x such that for any nonempty open subset U the set

$$N_U=\{n\in \mathbb {N}\cup \{0\} \,:\,\text {for which} \,T^nx\in U \}$$

has a positive lower density in \(\mathbb {N}\). In [11], it was proved that nonscalar convolution operators, studied by Godefroy-Shapiro, are frequently hypercyclic.

The proof of Theorem 1.1 refers to different cases for \(\lambda\) and the values of \(\phi\) at the origin, which share a similar flavor to recent studies on algebras of hypercyclic vectors for convolution operators by F. Bayart, J. Bès et al. [2, 8]. Specifically the proof of Theorem 1.1 breaks down into these partial cases.

  1. 1

    \(|\lambda |<1\), and   \(\lambda ^n=1\) for some \(n\in \mathbb {N};\)

  2. 2

    \(|\phi (0)|> 1\) and \(|\lambda |\ge 1;\)

  3. 3

    \(0<|\phi (0)|\le 1\) and \(|\lambda | >1;\)

  4. 4

    \(0<|\phi (0)|\le 1\) and \(|\lambda |= 1;\)

  5. 5

    \(\phi (0)=0\) and \(|\lambda |\ge 1.\)

When \(|\lambda |<1\), the operator \(R_\lambda \phi (D)\) is not hypercyclic; therefore, it is not frequently hypercyclic. When \(|\lambda |=1\) and \(\lambda\) is a root of the unity, the same argument of that in [5, Remark 3.5] provides that \(R_\lambda \phi (D)\) is frequently hypercyclic. In case 2: \(|\phi (0)|> 1\) and \(|\lambda |\ge 1;\) \(R_\lambda \phi (D)\) is hypercyclic, and from these ideas we can get that \(R_\lambda \phi (D)\) is frequently hypercyclic whenever \(|\phi (0)|>1\) and \(|\lambda |>1\) (see [5, Remark 4.2]). Frequent hypercyclicity for \(R_\lambda \phi (D)\) can be deduced from the proof of hypercyclicity for the case \(\phi (0)=0\) and \(|\lambda |\ge 1\) (see [5, Remark 7.2]).

The same ideas to show the hypercyclicity of \(R_\lambda \phi (D)\) in the case \(|\lambda |>1\) and \(0<|\phi (0)|\le 1\) (see [5, Proposition 5.2]) allow to show the frequent hypercyclicity of \(R_\lambda \phi (D)\) when \(|\lambda |=1\) and \(0<|\phi (0)|<1\) (see [5, Remark 5.4]). Moreover, although it is not specified in any comment in [5], with a little more work, frequent hypercyclicity for the case \(0<|\phi (0)|\le 1\) and \(|\lambda |>1\) could be obtained. In this paper, we will provide a new argument which proves this case.

As far as we know, the ideas of [5] cannot be adapted to obtain frequent hypercyclicity in the case \(|\phi (0)|=1\), \(|\lambda |=1\), and \(\lambda\) an irrational rotation. This is the most intriguing case. The ideas developed in [5] for the case \(\lambda =e^{2\pi i \theta }\) an irrational rotation and \(|\phi (0)|=1\) use an argument on normal families that, as far as we know, is not enough to get frequent hypercyclicity.

In this paper, we aim to present a new and direct proof of frequent hypercyclicity for \(R_\lambda \phi (D)\), which unifies several partial results in [5]. Moreover, in this paper, we fully characterize when an extended \(\lambda\)-eigenoperator is frequently hypercyclic, solving a question posed in [5, Problem 7.3].

When \(\phi\) has some zero a, then the operator \(R_\lambda \phi (D)\) is not injective, which means that there exist multiple right inverses for \(R_\lambda \phi (D)\) which can be defined on dense subsets and extended by continuity. The difficulty of such a question relies on this fact. Using the Pólya representation of an entire function, we get different ways to define right inverses of \(R_\lambda \phi (D)\) on a total set consisting of exponential functions.

In the same way that Godefroy-Shapiro’s result pivots on the multiples of the identity, our statement pivots on the multiples of another distinguished operator \(C_{\lambda ,b}f=f(\lambda z+b)\), \(\lambda \ne 1\), that is, composition operators induced by maps \(\lambda z+b\), \(\lambda \ne 1\), which are called by some authors affine endomorphisms of \(\mathbb {C}\) (see [6]).

Main theorem. Assume that \(\lambda \ne 1\). If T is an extended \(\lambda\)-eigenoperator of D, then T is frequently hypercyclic if and only if \(|\lambda |\ge 1\) and T is not a multiple of a composition operator induced by an affine endomorphism.

The paper is organized as follows. In the “Preliminaries” section, we recall the Frequent Hypercyclicity Criterion and the Pólya representation of an entire function which will be the main tools to prove our main result. In the “Main result” section, we will show that \(R_\lambda \phi (D)\) is frequently hypercyclic in the cases \(\phi (0)\ne 0\) and \(|\lambda |\ge 1\) which covers the case that remains unsolved and answers the Problem 7.3 in [5].

Preliminaries

Let \(\Vert \cdot \Vert\) denote an F-norm that induces the topology of a Fréchet space X. A series \(\sum _{n=1}^\infty x_n\) is unconditionally convergent (see [14, Appendix A]) if for any \(\varepsilon >0\) there is some \(N\in \mathbb {N}\) such that for any finite set \(F \subset \{N,N + 1,N + 2,...\}\) we have that

$$\left\| \sum _{n\in F}x_n\right\| <\varepsilon .$$

To get frequent hypercyclicity, we will use the following sufficient condition discovered by F. Bayart and S. Grivaux in [3]. We formulate this sufficient condition in a strengthened form according to A. Bonilla and K. G. Grosse-Erdmann [12, Theorem 2.1]:

Theorem 2.1

(Frequent Hypercyclicity Criterion) Let T be an operator on an F-space \(\mathcal {X}\). If there is a dense subset \(X_0\) and a map \(S:X_0\rightarrow X_0\) such that, for any \(x_0\in X_0\)

  1. 1

    \(\sum _{n=0}^\infty T^nx_0\) converges unconditionally.

  2. 2

    \(\sum _{n=0}^\infty S^nx_0\) converges unconditionally.

  3. 3

    \(TSx_0=x_0\).

Then, T is frequently hypercyclic.

Let f be an entire function of exponential type \(f(z)=\sum _{n=0}^\infty a_nz^n\). The Borel transform of f is defined as

$$Bf(z)=\sum _{n=0}^\infty \frac{n!a_n}{z^{n+1}}.$$

It is well known that Bf(z) is analytic on \(|z|>c\) for some \(c>0\). It is worthy to note that the type of f can be taken as such a constant c. In particular, for the exponentials \(f_b(z)=e^{bz}\), we have

$$\begin{aligned} Bf_b(z)= & {} \sum _{n=0}^\infty \frac{b^n}{z^{n+1}}\\= &\;{} \frac{1}{z} \frac{1}{1-\frac{b}{z}}=\frac{1}{z-b} \end{aligned}$$

which is analytic on \(|z|>|b|\).

Pólya representation of f (see [10, Theorem 5.3.5] p. 74) asserts that if Bf(z) is analytic on \(|z|>c\), then for any \(R>c\) we have:

$$f(z)=\frac{1}{2\pi i} \oint _{|t|=R} e^{zt} Bf(t)\,dt.$$

Thus, using the above formula, if \(\phi (D)=\sum _{n=0}^\infty \beta _n D^n\), then

$$\begin{aligned} R_\lambda \phi (D)f(z)= &\;{} R_\lambda \left( \sum _n \beta _n \frac{1}{2\pi i} \oint _{|t|=R} t^n e^{tz} Bf(t)\, dt\right) \\= &\;{} R_\lambda \frac{1}{2\pi i} \oint _{|t|=R} \left( \sum _n \beta _n t^n\right) e^{zt}Bf(t)\, dt\\=\;& {} \frac{1}{2\pi i} \oint _{|t|=R} \phi (t) e^{\lambda zt}Bf(t)\,dt. \end{aligned}$$

And iterating we get:

$$\left( R_\lambda \phi (D)\right) ^n f(z)=\frac{1}{2\pi i} \oint _{|t|=R} \phi (t) \phi (\lambda t)\cdots \phi (\lambda ^{n-1}t) e^{\lambda ^{n} zt} Bf(t)\,dt.$$

Set \(\omega =1/\lambda\). For some \(R>c\), defining

$$\begin{aligned} S_{1,\lambda }f(z)=\frac{1}{2\pi i} \oint _{|t|=R} \frac{1}{\phi (\omega t)} e^{\omega zt} Bf(t)\,dt, \end{aligned}$$
(1)

arguing as in the above computation of \(R_\lambda \phi (D)f(z)\) we get \(R_\lambda \phi (D)S_{1,\lambda }f=f\). Let us point out that \(R>c\) is chosen such that \(\phi (z)\) has no zeros on the circle \(|z|=R|\lambda |\), because the zeros of \(\phi\) are isolated. That is, the integral in Eq. (1) is well defined. Let us remark that Eq. (1) is a way to define a right inverse of \(R_\lambda \phi (D)\). Iterating we obtain a formula to define the right inverse of the powers of \(R_\lambda \phi (D)\):

$$\begin{aligned} S_{n,\lambda }f(z)=\frac{1}{2\pi i} \oint _{|t|=R} \frac{1}{\phi (\omega t)\cdots \phi (\omega ^nt)} e^{\omega ^n zt} Bf(t)\,dt, \end{aligned}$$
(2)

However, in general, when \(|\omega |\le 1\), the iterates above do not converge to zero. For instance, when \(|\omega |<1\), the denominator converges to \(\phi (0)\), and when \(|\phi (0)|<1\) these integrals converge to infinity. However, this equation provides an idea on how to construct a sequence of mappings \(S_{k,\lambda }\) satisfying the assumptions of Frequent Hypercyclicity Criterion.

Main result

To obtain convergence to zero, we will define the mappings \(S_{k,\lambda }\) by choosing some factor of \(\phi\) on the denominator.

Proposition 3.1

Assume that \(\phi ^{-1}\{0\}\ne \emptyset\) and \(\lambda \ne 1\). If \(\phi (0)\ne 0\) and \(|\lambda |\ge 1\), then \(T=R_\lambda \phi (D)\) is frequently hypercyclic.

Proof

Let a be some zero of \(\phi\) with a minimal absolute value and let us denote |a| by \(r_0\).

Since \(|\omega |\le 1\), the sequence \((\omega ^na)_n\) has an accumulation point in \(\mathbb {C}\); therefore, the subset \(X_0=\text {span}\, \{e^{\omega ^na z}\,:n\ge 0\}\) is dense in \(\mathcal {H}({\mathbb {C}})\).

On the other hand, since \(Te^{\omega ^naz}=R_\lambda \phi (D)e^{\omega ^naz}=\phi (\omega ^na)e^{\omega ^{n-1}az}\), we get that \(T^ke^{\omega ^naz}=0\) for \(k>n\). Therefore, if \(x_0\in X_0\), then \(T^nx_0=0\) for n large enough. Thus, if \(x_0\in X_0\), then

$$\sum _{n=0}^\infty T^nx_0$$

converges unconditionally because it consists of finite numbers of non zero summands.

The proof will be finished if we can define a sequence of mappings \(S_{k,\lambda }\) on the exponentials \(e_n(z)=e^{\omega ^naz}\) (\(n\ge 0\)) satisfying the following conditions:

  1. 1

    \(\sum _{k=1}^\infty S_{k,\lambda } e_n\) is unconditionally convergent.

  2. 2

    \((R_\lambda \phi (D))^kS_{k,\lambda } e_n=e_n\).

Let \(a_1=a\) be a zero of \(\phi\) of minimum modulus \(r_0=|a|\). Let us consider \(a_1=a, a_2,\cdots , a_m\) the zeros of \(\phi\) in the circle \(\partial (r_0\mathbb {D})\), repeated as its multiplicities indicate. If we denote

$$P(z)=\phi (0)\left( 1-\frac{z}{a_1}\right) \cdots \left( 1-\frac{z}{a_m}\right) ,$$

then the analytic function \(\xi (z)=\frac{\phi (z)}{P(z)}\) does not vanish on the disk \(\overline{r_0\mathbb {D}}\). Let us denote by \(0<M=\frac{1}{\min _{z\in r_0\mathbb {D}} |\xi (z)|} =\max _{z\in r_0\mathbb {D}} \frac{1}{|\xi (z)|}\).

Set \(M_0>M\). There exists \(R>r_0\), such that \(|P(z)|>M_0>M\) for \(|z|\ge R\). Let us define

$$\begin{aligned} (S_{k,\lambda } e_n)(z)=\frac{1}{\xi (\omega ^{n+1}a)\cdots \xi (\omega ^{n+k}a)}\frac{1}{2\pi i} \oint _{|t|=R} \frac{1}{P(\omega t)\cdots P(\omega ^kt)} e^{\omega ^kzt} \frac{1}{t-\omega ^na}\, dt. \end{aligned}$$
(3)

And we extend \(S_{k,\lambda }\) by linearity on \(X_0\). Now, let us observe that

$$\begin{aligned} |(S_{k,\lambda } e_n)(z)|\le & {}\;M^k \frac{1}{2\pi } 2 \pi R\, e^{R|z|} \frac{1}{M_0^k (R-|\omega ^na|)}\\\le &\;{} \left(\frac{M}{M_0}\right) ^k \frac{Re^{r|z|}}{R-|a|}\rightarrow 0 \end{aligned}$$

uniformly on compact subsets as \(k\rightarrow \infty\). Moreover, we get that the series

$$\sum _{k=1}^{\infty } S_{k,\lambda } e_n(z)$$

is unconditionally convergent.

Finally, by construction, we get:

$$\begin{aligned} R_\lambda \phi (D)S_{1,\lambda }e_n(z)= &\;{} \frac{1}{\xi (\omega ^{n+1}a)}R_\lambda \phi (D)\left( \frac{1}{2\pi i} \oint _{|z|=R}\frac{1}{P(\omega t)}e^{\omega zt} \frac{1}{t-\omega ^na}\,dt\right) \\= &\;{} \frac{1}{\xi (\omega ^{n+1}a)}\frac{1}{2\pi i} \oint _{|t|=R} \frac{\phi (\omega t)}{P(\omega t) }e^{zt} \frac{1}{t-\omega ^na}\,dt\\=\;& {} \frac{\xi (\omega ^{n+1}a)}{\xi (\omega ^{n+1}a)}e^{\omega ^naz}=e_n(z), \end{aligned}$$

where the last equality is just to apply the residue theorem.

Reasoning inductively we get:

$$\begin{aligned} (R_\lambda \phi (D))^kS_{k,\lambda }e_{n}(z)= &\;{} \frac{1}{\xi (\omega ^{n+1}a)\cdots \xi (\omega ^{n+k}a)}\frac{1}{2\pi i}\oint _{|t|=R} \xi (\omega t)\cdots \xi (\omega ^{k}t) e^{zt} \frac{1}{t-\omega ^na}\, dt.\\= &\;{} \frac{\xi (\omega ^{n+1}a)\cdots \xi (\omega ^{n+k}a)}{\xi (\omega ^{n+1}a)\cdots \xi (\omega ^{n+k}a)}e^{\omega ^naz}=e_{n}(z). \end{aligned}$$

Therefore, all requirements of the Frequent Hypercyclicity Criterion are satisfied. Hence, \(R_\lambda \phi (D)\) is frequently hypercyclic as we desired to prove.

Remark 3.1

Let us point out that the main theorem, stated in the “Introduction,” follows using Propositions 2.4 and 3.3 and Remark 7.2 in [5] and Proposition 3.1 above.