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Mean-boundedness and Littlewood-Paley for separation-preserving operators. (English) Zbl 0888.42004

Summary: Suppose that \((\Omega ,\mathcal{M},\mu)\) is a \(\sigma\)-finite measure space, \(1<p<\infty\), and \(T: L^{p}(\mu)\to L^{p}(\mu)\) is a bounded, invertible, separation-preserving linear operator such that the linear modulus of \(T\) is mean-bounded. We show that \(T\) has a spectral representation formally resembling that for a unitary operator, but involving a family of projections in \(L^{p}(\mu)\) which has weaker properties than those associated with a countably additive Borel spectral measure. This spectral decomposition for \(T\) is shown to produce a strongly countably additive spectral measure on the “dyadic sigma-algebra” of \(\mathbb{T}\), and to furnish \(L^{p}(\mu)\) with abstract analogues of the classical Littlewood-Paley and Vector-Valued M. Riesz Theorems for \(\ell^{p}(\mathbb{Z})\).

MSC:

42A45 Multipliers in one variable harmonic analysis
47B38 Linear operators on function spaces (general)
42B25 Maximal functions, Littlewood-Paley theory
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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