Joint and double coboundaries of commuting contractions

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 $\hat{I}\pm$,${\hat{I}}^2\in (0,1)$ are irrational, with $T_{\hat{I}}\pm$ and $T_{\hat{I}}^2$ 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 $L_1(\mathbb{T})$).