Extension to other sequence spaces

Extend the radial characterizations of Furstenberg-family transitivity for backward shifts on the infinite lattice graph G_infty from the weighted Hilbert space ell^2(V_infty, mu) to the weighted spaces ell^p(V_infty, mu), 1 <= p < infinity, and c_0(V_infty, mu).

Background

The paper obtains radial characterizations of Furstenberg-family transitivity on G_infty only in the weighted ell^2 setting. The proof relies on finite-dimensional transfer matrices and sharp smallest-singular-value estimates, which are intrinsically Hilbert-space methods.

The authors explicitly ask whether these characterizations can be extended to weighted ell^p and c_0 spaces. The stated obstacle is the lack of corresponding sharp right-inverse estimates in norms where the singular-value argument is not directly applicable.