Mean-boundedness and Littlewood-Paley for separation-preserving operators

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 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 $\ell ^{p}(\mathbb {Z})$.