Fréchet代数上允许超循环代数的算子的典型性
Typicality of operators on Fréchet algebras admitting a hypercyclic algebra
AI总结：
本文研究Fréchet代数上连续线性算子的典型性，证明在整函数代数和序列代数上，典型算子支持超循环代数。
摘要：
本文致力于研究在Fréchet代数上作用的某些连续线性算子类的典型性质（在Baire范畴意义上），这些算子空间赋有点态收敛拓扑。我们的主要结果表明，在作用于整函数代数$H(\mathbb{C})$（即$\mathbb{C}$上的整函数构成的代数）的连续算子的自然Polish空间中，典型算子支持一个超循环代数。我们还研究了复Fréchet代数$X=\ell_{p}(\mathbb{N})$（$1\le p<+\infty$）或$X=c_{0}(\mathbb{N})$（配备坐标乘积）的情形，并证明当$M>1$时，$X$上范数小于或等于$M$的典型算子允许一个超循环代数。
英文摘要：
This paper is devoted to the study of typical properties (in the Baire Category sense) of certain classes of continuous linear operators acting on Fréchet algebras, endowed with the topology of pointwise convergence. Our main results show that within natural Polish spaces of continuous operators acting on the algebra $H(\mathbb{C})$ of entire functions on $\mathbb{C}$, a typical operator supports a hypercyclic algebra. We also investigate the case of the complex Fréchet algebras $X=\ell_{p}(\mathbb{N})$, $1\le p<+\infty$, or $X=c_{0}(\mathbb{N})$ endowed with the coordinatewise product, and show that whenever $M>1$, a typical operator on $X$ of norm less than or equal to $M$ admits a hypercyclic algebra.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
