Joint and double coboundaries of commuting contractions

Let T and S be commuting contractions on a Banach space X. The elements of (I-T)(I-S)X are called double coboundaries, and the elements of (I-T)X∩(I-S)X are called joint coboundaries. For U and V the unitary operators induced on L2 by commuting invertible measure-preserving transformations which generate an aperiodic Z2-action, we show that there are joint coboundaries in L2 which are not double coboundaries. We prove that if α,β ∈ (0, 1) are irrational, with Tα and Tβ induced on L1(T) by the corresponding rotations, then there are joint coboundaries in C(T) which are not measurable double co-boundaries (hence not double co-boundaries in L1(T)).