| For each outer function O in the Smirnov class N + and each p ∈ (0, infinity), we define a subspace NpW ⊂ Hp that carries an operation analogous to complex conjugation. We relate this operation to the inner-outer factorization of functions in the NpW spaces. If B denotes the backward shift, then each B-invariant subspace of Hp for p ∈ [1, infinity) is a NpW space and every B-invariant subspace of Hp for p ∈ (0, 1) is contained in a NpW space. For p ∈ (0, infinity), we obtain an explicit representation of noncyclic functions in Hp and we characterize the generators of the B-invariant subspaces of Hp for p ∈ [1, infinity). Applications include a concrete description of all H 2 functions that are pseudocontinuable of bounded type, a solution to the 2 x 2 scalar-valued Darlington Synthesis problem, and a parameterization of all 2 x 2 matrices with Hinfinity entries that are unitary almost everywhere on the unit circle. For p ∈ (1, infinity), we identify each nontrivial Toeplitz kernel with a NpW space and relate this result to representation theorems of Hayashi and Dyakonov. |
