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.
展开▼
机译：对于Smirnov类N +中的每个外部函数O和每个p∈（0，无穷大），我们定义一个子空间NpW⊂Hp，该子空间进行类似于复共轭的运算。我们将此操作与NpW空间中函数的内外分解相关。如果B表示向后移动，则p∈[1，infinity）的Hp的每个B不变子空间是一个NpW空间，p∈（0，1）的Hp的每个B不变子空间都包含在NpW空间中。对于p∈（0，无穷大），我们获得了Hp中非循环函数的显式表示，并表征了p∈[1，无穷大]的Hp B不变子空间的生成器。应用包括对所有H 2函数的具体描述，这些H 2函数是有界类型的伪连续的，是2 x 2标量值达林顿综合问题的解决方案，并且所有2 x 2矩阵的参数化都带有Hinfinity项，几乎在所有位置上都是unit性的单位圆。对于p∈（1，无穷大），我们用NpW空间标识每个非平凡的Toeplitz核，并将该结果与Hayashi和Dyakonov的表示定理联系起来。
展开▼
