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Rate of growth of frequently hypercyclic functions. (English) Zbl 1230.47019

A (continuous and linear) operator \(T\) on a topological vector space \(X\) is said to be hypercyclic (resp. frequently hypercyclic) if there exists a vector \(x \in X\) such that for every non-empty open subset \(\mathcal{U}\) of \(X\) there is an \(n_0 \in \mathbb{N}\) (resp. a strictly increasing sequence \((n_k)\subset \mathbb{N}\)) with \(T^{n_0}x \in \mathcal{U}\) (resp. \((\frac{n_k}{k})\) is bounded and \(T^{n_k}x \in \mathcal{U}\) for all \(k \in \mathbb{N}\)).
In 1929 Birkhoff showed that the translation operators \(T_a f(z):= f(z+a)\) (\(a \neq 0\)) on the space \(H(\mathbb{C})\) of entire functions are hypercyclic. In 1952 MacLane obtained the same for the differentiation operator \(D: H(\mathbb{C}) \to H(\mathbb{C})\), \(Df(z):= f'(z)\). A natural problem is to ask how slowly a hypercyclic function can grow at \(\infty\). Several authors have widely studied this question. An optimal result for the operator \(D\) was obtained in [K.-G. Grosse-Erdmann, Complex Variables Theory Appl. 15, 193–196 (1990; Zbl 0682.30021)] and, independently, in [S. A. Shkarin, Mosc. Univ. Math. Bull. 48, 49–51 (1993; Zbl 0845.30015)]: If \(\varphi : \mathbb{R}_+ \to \mathbb{R}_+\) is any function with \(\varphi(r) \to \infty \) as \(r \to \infty\), then there is a \(D\)-hypercyclic entire function \(f\) with \[ |f(z)| \leq \varphi (r) \frac{e^r}{\sqrt{r}} \;for \;|z|=r\, \text{ sufficiently \;large}; \] however, there is no \(D\)-hypercyclic entire function with \[ |f(z)| \leq C \frac{e^r}{\sqrt{r}} \;for \;|z|=r>0, \text{ where }\;C>0. \] In the main result of this paper the authors give growth rates for which \(D\)-frequently hypercyclic functions exist.
Main result: Let \(1 \leq p \leq \infty\) and set \(a=1/2 \max \{2, p \}\), \(b = 1/2 \min \{2, p \}\). Then,
(a) For any function \(\varphi : \mathbb{R}_+ \to \mathbb{R}_+\) with \(\varphi(r) \to \infty \) as \(r \to \infty\), there is a \(D\)-frequently hypercyclic entire function \(f\) with \[ M_p (f,r) \leq \varphi (r) \frac{e^r}{r^a} \;for \;|z|=r \text{ sufficiently \;large}; \] (b) For any function \(\psi : \mathbb{R}_+ \to \mathbb{R}_+\) with \(\psi (r) \to 0 \) as \(r \to \infty\), there is no a \(D\)-frequently hypercyclic entire function \(f\) with \[ M_p (f,r) \leq \psi (r) \frac{e^r}{r^a} \;for \;|z|=r \text{ sufficiently \;large}; \] (Here \(M_p(f,r)= (\frac{1}{2\pi} \int_0^{2\pi} |f(re^{it})|^p dt)^{1/p}\) if \(1 \leq p <\infty\) and \(M_\infty (f,r) = \sup_{|z|=r} |f(z)|\).)
For the translation operator \(T_a\) (\(a\neq 0\)), the authors show that there are \(T_a\)-frequently hypercyclic entire functions of order 1 and any given positive type, but not of type 0. The positive part was also obtained, by a different method, in [A. Bonilla and K.-G. Grosse-Erdmann, Integral Equations Oper. Theory 56, No. 2, 151–162 (2006; Zbl 1114.47004)].
Finally, the authors prove the existence of frequently hypercyclic harmonic functions for the translation operator and they study the rate of growth of harmonic functions that are frequently hypercyclic for partial differentiation operators.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
30D15 Special classes of entire functions of one complex variable and growth estimates
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B38 Linear operators on function spaces (general)
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