AI 中文总结
该论文在Fréchet空间、整函数空间及有限维赋范空间乘积上，证明了加权复合算子、双侧加权前向移位算子的跟踪性质，以及乘积空间上线性算子无度量扩张性，解答了相关公开问题。
AI 中文摘要
我们证明：(1) 当0<|c|<1且B∈GL_d(ℂ)的谱半径小于1时，Fréchet空间H(ℂ^d)（ℂ^d上整函数构成的空间，赋予紧开拓扑）上的加权复合算子Tf(z)=c f(Bz)具有跟踪性质；(2) ℤ上的双侧加权前向移位算子，只要权重非零，就具有跟踪性质；(3) 可数无限个非零有限维赋范空间的乘积上，任意相容度量下的连续线性算子均无度量正扩张性，任意相容度量下的线性同胚均无度量扩张性。这些结果回答了文献[BCDFP]中问题A的H(ℂ)部分与问题B的ℤ部分。
英文摘要
We prove that (1) the weighted composition operator \(Tf(z)=c f(Bz)\) on the Fréchet space \(H(\mathbb{C}^d)\) of entire functions on \(\mathbb{C}^d\) (\(d\in\mathbb{N}\)) with the compact-open topology has the shadowing property whenever \(0<|c|<1\) and \(B\in\operatorname{GL}_d(\mathbb C)\) has spectral radius less than \(1\); (2) every bilateral weighted forward shift on \(\mathbb K^{\mathbb Z}\) with nonzero weights has the shadowing property; and (3) no continuous linear operator on a countably infinite product of nonzero finite-dimensional normed spaces is metrically positively expansive for any compatible metric, and no linear homeomorphism on such a product is metrically expansive for any compatible metric. These results answer the \(H(\mathbb C)\)-part of \cite[Problem A]{BCDFP} and the \(\mathbb K^{\mathbb Z}\)-part of \cite[Problem B]{BCDFP}.
