MEAN-BOUNDEDNESS AND LITTLEWOOD-PALEY FOR SEPARATION-PRESERVING OPERATORS

Suppose that (Omega, M, mu) is a sigma-finite measure space, 1 < p < i nfinity, and T: L(p)(mu) --> L(p)(mu) is a bounded, invertible, separa tion-preserving linear operator such that the linear modulus of T is m ean-bounded. We show that T has a spectral representation formally res embling that for a unitary operator, but involving a family of project ions in L(p)(mu) which has weaker properties than those associated wit h a countably additive Borel spectral measure. This spectral decomposi tion for T is shown to produce a strongly countably spectral measure o n the ''dyadic sigma-algebra'' of T, and to furnish L(p)(mu) with abst ract analogues of the classical Littlewood-Paley and Vector-Valued M R iesz Theorems for l(p) (Z).