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Multiplication Invariant Subspaces of Hardy Spaces

Published online by Cambridge University Press:  20 November 2018

T. L. Lance  and
M. I. Stessin
T. L. Lance
Affiliation:
Department of Mathematics and Statistics, University at Albany, State University of New York, Albany, NY, USA 12222 e-mail: lance@math.albany.edu, stessin@math.albany.edu
M. I. Stessin
Affiliation:
Department of Mathematics and Statistics, University at Albany, State University of New York, Albany, NY, USA 12222 e-mail: lance@math.albany.edu, stessin@math.albany.edu
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Abstract

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This 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.

Keywords

30H0546E1547B38
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 1 , 01 February 1997 , pp. 100 - 118
Copyright
Copyright © Canadian Mathematical Society 1997

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