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Multiplicative structures of hypercyclic functions for convolution operators. (English) Zbl 1424.47020

In this paper, \(\mathcal{H}(\mathbb{C})\) denotes all entire funtions \(\mathbb{C} \to \mathbb{C}\) with the topology of uniform convergence on the compact subsets of \(\mathbb{C}\); this makes it a topological vector space. Let \(L(\mathcal{H}(\mathbb{C}))\) stand for all linear continuous self maps \(T: \mathcal{H}(\mathbb{C}) \to \mathcal{H} (\mathbb{C})\), let \(e^{\mathcal{H}(\mathbb{C})}\) denote the family of non-vanishing entire functions and \(e_{+,0}^{\mathcal{H}(\mathbb{C})} = \{f \in e^{\mathcal{H}(\mathbb{C})}: f(0) >0 \}.\) An operator \(T \in L(\mathcal{H}(\mathbb{C})) \) is called a convolution operator if it commutes with its translation, and is called hypercyclic if there is an \(f \in \mathcal{H}(\mathbb{C})\) such that the orbit of \( \{ T^{n}(f): n \in \mathbb{N} \}\) is dense in \(\mathcal{H}(\mathbb{C})\); then \(f\) is called \(T\)-hypercyclic and the set of all these \(f \in \mathcal{H}(\mathbb{C})\) is denoted by \(HC(T)\). With a non-constant entire function \(\Phi(z) = \sum_{n \geq 0} a_{n} z^{n}\), an operator \(T \in L(\mathcal{H}(\mathbb{C}))\) defined by \(T(f) = \sum_{n \geq 0}a_{n} f^{n}\) is associated. In this paper, the authors deal with the size of \(HC(T)\) from algebraic and from certain differential operators points of view. After proving many preliminary results, the authors prove their main result: Let \(\Phi\) be a non-constant entire function of subexponential type and \(\Phi(D)\) be the convolution operator associated with \(\Phi\). Then there is an infinitely generated multiplicatice group \(\mathcal{G} \subset \mathcal{H}(\mathbb{C})\), having each of its element, except the function 1, \(\Phi(D)\)-hypercyclic. Further, \(\mathcal{G}\) is a dense subgroup of \(e_{+,0}^{\mathcal{H}(\mathbb{C})}\) and the algebra generated by \(\mathcal{G}\) is infinitely generated. This result is further extended to simply connected domains \(G \subset \mathbb{C}\) under some special composition operators.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
46E10 Topological linear spaces of continuous, differentiable or analytic functions
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