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Joint and double coboundaries of commuting contractions. (English) Zbl 07403323

Summary: 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 \((I-T)X\cap(I-S)X\) are called joint coboundaries. 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 co-boundaries (hence not double co-boundaries in \(L_1(\mathbb{T}))\).

MSC:

47A35 Ergodic theory of linear operators
47A10 Spectrum, resolvent
37A05 Dynamical aspects of measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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