两族超循环非卷积算子

A. Myers, Muhammadyusuf Odinaev, David Walmsley
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引用次数: 0

摘要

设$H(\mathbb{C})$为紧集上具有一致收敛拓扑的所有完整函数的集合。设$\lambda,b\in\mathbb{C}$, $C_\gamma:H(\mathbb{C})\to H(\mathbb{C})$是复合运算符$C_\gamma f(z)=f(\lambda z+b)$, $D$是导数运算符。我们扩展了关于非卷积算子$T_{\lambda,b}=C_\gamma \circ D$的超循环性的结果,证明当$|\lambda|\geq 1$时,算子的代数\begin{align*} \{\psi(T_{\lambda,b}): \psi(z)\in H(\mathbb{C}), \psi(0)=0 \text{ and } \psi(T_{\lambda,b}) \text{ is continuous}\} \end{align*}和算子族\begin{align*} \{C_\gamma\circ\varphi(D): \varphi(z) \text{ is an entire function of exponential type with } \varphi(0)=0\} \end{align*}完全由超循环算子组成(即每个算子都有一个密集的轨道)。
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Two families of hypercyclic nonconvolution operators
Let $H(\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $\lambda,b\in\mathbb{C}$, let $C_\gamma:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator $C_\gamma f(z)=f(\lambda z+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{\lambda,b}=C_\gamma \circ D$ by showing that whenever $|\lambda|\geq 1$, the algebra of operators \begin{align*} \{\psi(T_{\lambda,b}): \psi(z)\in H(\mathbb{C}), \psi(0)=0 \text{ and } \psi(T_{\lambda,b}) \text{ is continuous}\} \end{align*} and the family of operators \begin{align*} \{C_\gamma\circ\varphi(D): \varphi(z) \text{ is an entire function of exponential type with } \varphi(0)=0\} \end{align*} consist entirely of hypercyclic operators (i.e., each operator has a dense orbit).
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