Shadowing and metric expansivity on Fréchet spaces

We prove that (1) the weighted composition operator $Tf(z)=c f(Bz)$ on the Fr\'echet 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}.