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 a unilateral shift on Hilbert space, I identity operator and x given vector. The are expressed terms of Wold-type decomposition determined by growth iterates at x. We revisit prove following result. Let be an isometry acting space $$\mathcal H$$ let $$x \in \mathcal . Suppose that $$ \sum _{k=0}^\infty k \Vert T^{*k} < \infty Then range $$(I-T)$$ if (and only if) $$\left\Vert\sum _{k= 0}^n T^k \right\Vert o(\sqrt{n}).$$ When merely contraction, additional assumption. Some applications to $$L^2$$ -solutions functional $$f(x)-f(2x) F(x)$$ , considered Fortet Kac, given.