Abstract

A differentiation-invariant weighted Fréchet space \({\mathcal E}(\varphi)\) of infinitely differentiable functions in \({\mathbb R}^n\) generated by a countable family \(\varphi\) of continuous real-valued functions in \({\mathbb R}^n\) is considered. It is shown that, under minimal restrictions on \(\varphi\), any continuous linear operator on \({\mathcal E}(\varphi)\) that is not a scalar multiple of the identity mapping and commutes with the partial differentiation operators is hypercyclic. Examples of hypercyclic operators in \({\mathcal E}(\varphi)\) are presented for cases in which the space \({\mathcal E}(\varphi)\) is translation invariant.

中文翻译：

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无限可微函数加权空间中的超循环算子

摘要

由连续实数的可数族\(\varphi\)生成的\({\mathbb R}^n\)中无限可微函数的微分不变加权Fréchet空间\({\mathcal E}(\varphi)\)考虑\({\mathbb R}^n\)中的-值函数。结果表明，在对\(\varphi\)进行最小限制的情况下， \({\mathcal E}(\varphi)\)上的任何连续线性算子不是恒等映射的标量倍数，并且与偏微分交换算子是超循环的。\({\mathcal E}(\varphi)\)中的超循环运算符示例针对空间\({\mathcal E}(\varphi)\) 的情况提供是平移不变的。