Joint and double coboundaries of transformations  an application of maximal spectral type of spectral measures

Let T be a bounded linear operator on a Banach space X; the elements of (I − T)X are called coboundaries. For two commuting operators T and S, elements of (I − T)X ∩ (I − S)X are called joint coboundaries, and those of (I − T)(I − S)X are double coboundaries. By commutativity, double coboundaries are joint ones. Are there any other? Let θ and τ be commuting invertible measure preserving transformations of (Ω, Σ, m), with corresponding unitary operators induced on L2(m). We prove the existence of a joint coboundary g ∈ (I − U)L2 ∩ (I − V )L2 which is not in (I − U)(I − V )L2. For the proof, let E be the spectral measure on T 2 obtained by Stone’s spectral theorem. Joint and double coboundaries are characterized using E, and properties of the maximal spectral type of E, together with a result of Foia³ on multiplicative spectral measures acting on L2, are used to show the existence of the required function.