Rochberg's abstract coboundary theorem revisited

TL;DR

This paper revisits Rochberg's coboundary theorem, providing new conditions for the solvability of the equation $(I-T)y = x$ in Hilbert spaces, with applications to classical functional equations.

Contribution

It extends Rochberg's theorem to isometries and contractions, establishing new criteria involving growth conditions and sum convergence for the existence of solutions.

Findings

Established new conditions for $(I-T)y = x$ solvability for isometries.

Extended results to contractions with additional assumptions.

Applied findings to classical functional equations like $f(x)-f(2x) = F(x)$.

Abstract

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.