ROCHBERG’S ABSTRACT COBOUNDARY THEOREM REVISITED

The paper proves Theorem 1.4: for an isometry T and x with ∑_{k≥0} k ||T^{*k}x|| < ∞, x is a coboundary (x = (I−T)y) iff (1/n)||∑_{k=0}^{n} T^k x||^2 → 0. The proof reduces via Wold decomposition to the shift part and constructs y = ∑_r y_r with y_r := ∑_{k=0}^r T^k x_{r−k}, then shows ∑_r ||y_r||^2 < ∞ using two lemmas (a limit identity under ∑ ||T^{*j}u|| < ∞ and a projection/Fejér-type identity), which together yield the required estimates in terms of ∑ k ||T^{*k}x|| (the given assumption) . By contrast, the model’s Phase 2 argument leaves critical gaps: (i) it asserts a Hilbert-valued Hardy inequality to uniformly bound the Fejér approximants y^(N) but does not provide a valid bound under the paper’s hypothesis, and it incorrectly treats sup_n (1/n)||S_n x||^2 as uniformly bounded merely from its limit 0 (limit→0 does not imply bounded sup); (ii) it tries to control ∑_k k||d_k||^2 from ∑_k k||T^{*k}x||, but the needed implication (essentially ∑ ||T^{*k}x||^2 < ∞) is not justified under the paper’s weaker assumption. The paper’s construction avoids these pitfalls by a different decomposition-and-estimate route. Therefore the paper is correct; the model’s solution is incomplete/incorrect on key estimates.