Rochberg's abstract coboundary theorem revisited

Rochberg's coboundary theorem provides conditions under which the equation $(I-T)y = x$ is solvable in $y$. Here $T$ is a unilateral shift on Hilbert space, $I$ is the identity operator and $x$ is a given vector. The conditions are expressed in terms of Wold-type decomposition determined by $T$ and growth of iterates of $T$ at $x$. We revisit Rochberg's theorem and prove the following result. Let $T$ be an isometry acting on a Hilbert space $\mathcal{H}$ and let $x \in \mathcal{H}$. Suppose that $\sum_{k=0}^\infty k \| T^{*k} x \| < \infty$. Then $x$ is in the range of $(I-T)$ if (and only if) $\|\sum_{k= 0}^n T^k x \| = o(\sqrt{n}).$ When $T$ is merely a contraction, $x$ is a coboundary under an additional assumption. Some applications to $L^2$-solutions of the functional equation $f(x)-f(2x) = F(x)$, considered by Fortet and Kac, are given.