Characterization for non-radial weights

Characterize Furstenberg-family transitivity of bounded backward shifts on the weighted space ell^2(V_infty, mu) under general, non-radial weights mu by explicit conditions on the weight.

Background

The main infinite-lattice theorem assumes radial weights of the form mu_{i,j} = omega_{i+j}. This assumption makes the transfer operator between diagonal levels a scalar multiple of a Pascal-type matrix, enabling sharp right-inverse estimates.

For general non-radial weights, weights vary within each diagonal level, so the transfer matrices are no longer scalar multiples of the Pascal-type matrices. The authors explicitly identify controlling this within-diagonal variation as the central difficulty.