Frequently hypercyclic sequences of differential operators on the space of entire functions

Abstract: A criterion to obtain frequent hypercyclicity for a sequence of convolution operators on the space of entire functions on the complex plane is provided. The criterion involves that the generating functions of the operators do not vanish on an appropriate annulus, in the boundary of which the modulus of each term of the sequence is in some sense controlled by the preceding ones or the following ones.

• The paper establishes a sufficient frequent-hypercyclicity criterion for convolution-operator sequences on H(C), combining zero-freeness on an annulus, modulus comparisons, and reciprocal-decay summability.
• The method constructs right inverses with the Borel transform and applies the Bonilla–Grosse-Erdmann criterion, yielding frequent hypercyclicity for weighted powers such as (c_nΦ(D)^n) and non-power sequences.
• The results include (c_nD^n) under bounded positive successive-ratio growth and examples of infinite-order and genuinely non-autonomous operators, while identifying unresolved weaker growth conditions.

Context and motivation

The paper studies frequent hypercyclicity for sequences of convolution operators on the Fréchet space $H(\mathbb{C})$ of entire functions with the compact-open topology. The operators considered are of the form $\Phi(D)$, where $\Phi$ is an entire function of exponential type and $D$ denotes complex differentiation; by the Godefroy–Shapiro characterization, these are precisely the continuous linear operators on $H(\mathbb{C})$ commuting with all translations. While hypercyclicity of such sequences was analyzed earlier (notably in Bernal-González's 1999 work), the frequent variant — introduced by Bayart–Grivaux — had been established only for iterates of a single operator: Bonilla and Grosse-Erdmann showed that every non-scalar $\Phi(D)$ is frequently hypercyclic. The present note extends this to genuine sequences $(\Phi_n(D))$ whose members need not be iterates of one operator.

The main criterion

The central result is a sufficient condition for frequent hypercyclicity of $(\Phi_n(D))$, where each $\Phi_n \in \mathcal{E}$ (exponential type) and $\Phi_0 \equiv 1$. The hypothesis is geometric-comparative: there exist radii $\Phi(D)$0 and coefficient arrays $\Phi(D)$1 ($\Phi(D)$2), $\Phi(D)$3 ($\Phi(D)$4) such that:

• Uniform summability: $\Phi(D)$5 and $\Phi(D)$6;
• Zero-free annulus: $\Phi(D)$7 on the closed annulus $\Phi(D)$8 for every $\Phi(D)$9;
• Backward control: $\Phi$0 on $\Phi$1 for $\Phi$2;
• Forward control: $\Phi$3 on $\Phi$4 for $\Phi$5;
• Decay: $\Phi$6.

Under these conditions, $\Phi$7 is frequently hypercyclic on $\Phi$8.

The proof applies the Bonilla–Grosse-Erdmann frequent hypercyclicity criterion on F-spaces, with $\Phi$9, $D$0 the polynomials, and right inverses $D$1 constructed via the Borel transform. Specifically, since polynomials have exponential type zero, Pólya's representation gives $D$2 for any $D$3, and the zero-free condition allows

$D$4

which is well defined independently of $D$5 by Cauchy's theorem. Then $D$6 on $D$7 exactly, and the three unconditional-convergence conditions follow from contour estimates on the circles $D$8 combined with the summability assumptions. A notable feature is that condition (iv) holds with equality rather than merely asymptotically.

The authors concede that conditions (a)–(e) may appear ad hoc, tailored to fit the Bonilla–Grosse-Erdmann template; the second theorem addresses this concern by showing they are verifiable in concrete situations.

Applications to weighted powers

The main corollary concerns sequences $D$9 with nonzero scalars satisfying

$H(\mathbb{C})$0

If there exist circles $H(\mathbb{C})$1, $H(\mathbb{C})$2 on which $H(\mathbb{C})$3 and $H(\mathbb{C})$4 respectively, with $H(\mathbb{C})$5 zero-free on the intervening annulus, then $H(\mathbb{C})$6 is frequently hypercyclic. Taking $H(\mathbb{C})$7 yields that $H(\mathbb{C})$8 is frequently hypercyclic whenever the ratio condition holds — a family of finite-order examples. An infinite-order example is also given: $H(\mathbb{C})$9, where $\Phi(D)$0 is unit translation; here $\Phi(D)$1, and Rouché's theorem verifies the hypotheses on $\Phi(D)$2 and $\Phi(D)$3.

A further example exhibits a frequently hypercyclic sequence not of the form $\Phi(D)$4 at all: $\Phi(D)$5, with generating functions $\Phi(D)$6 shown zero-free on $\Phi(D)$7 via Rouché, and pairwise comparisons $\Phi(D)$8 on the inner circle and the reverse inequality on the outer circle. This demonstrates that the criterion genuinely covers sequences beyond scalar multiples of operator powers.

Limitations and open questions

Three questions are posed explicitly. First, the ratio condition on $\Phi(D)$9 implies but is strictly stronger than boundedness away from $(\Phi_n(D))$0 and $(\Phi_n(D))$1 of $(\Phi_n(D))$2; whether the latter alone suffices for frequent hypercyclicity of $(\Phi_n(D))$3 remains open, in contrast with the known hypercyclicity result under that weaker hypothesis (Bernal-González–Prado-Tendero). Second, hypercyclicity of $(\Phi_n(D))$4 is characterized by unboundedness of $(\Phi_n(D))$5, which is insufficient for frequent hypercyclicity since the large terms may occupy a set of full density complement; the paper asks whether $(\Phi_n(D))$6 implies frequent hypercyclicity. Third, an analogous frequent-hypercyclicity theory for operators $(\Phi_n(D))$7 with holomorphic coefficients $(\Phi_n(D))$8, studied previously for ordinary hypercyclicity, is not developed here. The criterion itself is sufficient only; no necessity or converse is claimed.

Conclusion

The paper supplies a workable sufficient criterion — phrased as zero-freeness plus comparative modulus control on an annulus — for frequent hypercyclicity of arbitrary sequences of convolution operators on $(\Phi_n(D))$9, together with concrete applications covering weighted powers of differential operators and a genuinely non-power sequence. It thereby extends the Bonilla–Grosse-Erdmann frequent hypercyclicity theorem from single operators to broad classes of non-autonomous sequences, while leaving the sharpness of the scalar-growth hypotheses as explicit open problems.