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

Suppose that (Omega, M, mu) is a sigma-finite measure space, 1 < p < infinity, and T: L(p)(mu) --> 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 T, and to furnish L(p)(mu) with abstract analogues of the classical Littlewood-Paley and Vector-Valued M Riesz Theorems for l(p) (Z).