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