Let Γ be a dense subgroup
of the real line R. Endow Γ with the discrete topology, and let K be the dual group
of Γ. Helson’s classic theory uses the spectral representation in Stone’s Theorem for
unitary groups to establish and implement a one-to-one correspondence Φ2
between the cocycles on K and the normalized simply invariant subspaces of
L2(K). Using our recent extension of Stone’s Theorem to UMD spaces, we
generalize Helson’s theory to Lp(K),1 < p < ∞, by producing spectral
decompositions of Lp(K) which provide a correspondence analogous to Φ2.
In particular this approach shows that every normalized simply invariant
subspace of Lp(K) is the range of a bounded idempotent. However, unlike the
situation in the L2-setting, our spectral decompositions do not stem from a
projection-valued measure. Instead they owe their origins to the Hilbert transform of
Lp(R). In the context of abstract UMD spaces, we develop the relationships
between holomorphic semigroup extensions and the spectral decompositions of
bounded one-parameter groups. The results are then applied to describe, in
terms of generalized analyticity, the normalized simply invariant subspaces of
Lp(K).
