Exotic operators on the Hardy spaces of the infinite-dimensional torus

We quantify the failure of interpolation between Hardy spaces $H^p({\mathbb{T}}^{\mathrm{\infty }})$ on the infinite-dimensional torus. For any set $A\subseteq [1,\mathrm{\infty }]$ such that $A\setminus \{\mathrm{\infty }\}$ is closed in $[1,\mathrm{\infty })$, we show that there exists an operator that is densely defined on $H^p({\mathbb{T}}^{\mathrm{\infty }})$ for all $1\le p<\mathrm{\infty }$ and weak-$\ast$ densely defined on $H^{\mathrm{\infty }}({\mathbb{T}}^{\mathrm{\infty }})$ that extends to a bounded linear operator on $H^p({\mathbb{T}}^{\mathrm{\infty }})$ if and only if $p\in A$.