Shadowing and metric expansivity on Fr\'echet spaces

We prove that (1) the weighted composition operator $Tf(z)=cf(Bz)$ 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 $0<\mathrm{\mid }c\mathrm{\mid }<1$ and $B\in {\mathrm{GL}}_d(\mathbb{C})$ 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}.