Joint and double coboundaries of commuting contractions

TL;DR

This paper investigates the properties of joint and double coboundaries for commuting contractions on Banach spaces, providing examples where joint coboundaries are not double coboundaries, with implications in ergodic theory and functional analysis.

Contribution

It demonstrates the existence of joint coboundaries that are not double coboundaries in specific settings involving measure-preserving transformations and rotations.

Findings

Existence of joint coboundaries not being double coboundaries in $L_2$ for certain transformations.

Existence of joint coboundaries in $C(\mathbb T)$ that are not measurable double coboundaries.

Counterexamples in ergodic theory showing differences between joint and double coboundaries.

Abstract

Let $T$ and $S$ be commuting contractions on a Banach space $X$. The elements of $(I-T)(I-S)X$ are called {\it double coboundaries}, and the elements of $(I-T)X \cap (I-S)X$ are called {\it joint cobundaries}. For $U$ and $V$ the unitary operators induced on $L_2$ by commuting invertible measure preserving transformations which generate an aperiodic $\mathbb Z^2$-action, we show that there are joint coboundaries in $L_2$ which are not double coboundaries. We prove that if $\alpha$,$\beta \in (0,1)$ are irrational, with $T_\alpha$ and $T_\beta$ induced on $L_1(\mathbb T)$ by the corresponding rotations, then there are joint coboundaries in $C(\mathbb T)$ which are not measurable double cobundaries (hence not double coboundaries in $L_1(\mathbb T)$).