无穷多复变数整函数空间上卷积算子的混沌
Chaos in convolution operators on the space of entire functions of infinitely many complex variables
摘要：
Goalry和Shapiro的一个经典结果指出，多复变数整函数空间$\mathcal{H}(\mathbb{C}^n)$上的每个非平凡卷积算子都是超循环的.与这个结果形成鲜明对比的是，Fávaro和Mujica证明了在无穷多个复变数的整函数的空间$\mathcal{H}(\mathbb{C}^\mathbb{N})$上没有平移算子是超循环的。本文研究了$\mathcal{H}(\mathbb{C}^\mathbb{N})$上卷积算子的线性动力学。首先证明了对于任意正整数$n$，$\mathcal{H}(\mathbb{C}^\mathbb{N})$上的卷积算子都不是循环的，也不是$n$ -超循环的.在不可度量化的拓扑向量空间中研究了Li-Yorke混沌的概念，证明了$\mathcal{H}(\mathbb{C}^\mathbb{N})$上的每个非平凡卷积算子都是Li-Yorke混沌的.
英文摘要：
A classical result of Godefroy and Shapiro states that every nontrivial convolution operator on the space $\mathcal{H}(\mathbb{C}^n)$ of entire functions of several complex variables is hypercyclic. In sharp contrast with this result Fávaro and Mujica show that no translation operator on the space $\mathcal{H}(\mathbb{C}^\mathbb{N})$ of entire functions of infinitely many complex variables is hypercyclic. In this work we study the linear dynamics of convolution operators on $\mathcal{H}(\mathbb{C}^\mathbb{N})$. First we show that no convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ is neither cyclic nor $n$-supercyclic for any positive integer $n$. After we study the notion of Li--Yorke chaos in non-metrizable topological vector spaces and we show that every nontrivial convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ is Li--Yorke chaotic.
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