Multiplication Invariant Subspaces of Hardy Spaces

Abstract

AbstractThis paper studies closed subspaces L of the Hardy spaces Hp which are g-invariant (i.e., g. L ⊆ L) where g is inner, g ≠ 1. If p = 2, theWold decomposition theorem implies that there is a countable “g-basis” f1, f2, . . . of L in the sense that L is a direct sum of spaces fj . H2[g] where H2[g] = {f o g | f ∈ H2}. The basis elements fj satisfy the additional property that ∫T |fj|2gk = 0, k = 1, 2, . . . . We call such functions g-2-inner. It also follows that any f ∈ H2 can be factored f = hf ,2 . (F2 o g) where hf,2 is g-2-inner and F is outer, generalizing the classical Riesz factorization. Using Lp estimates for the canonical decomposition of H2,we find a factorization f = hf ,p.(Fpog) for f ∈ Hp. If p ≤ 1 and g is a finite Blaschke product we obtain, for any g-invariant L ⊆ Hp, a finite g-basis of g-p-inner functions.