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Nearly invariant and the backward shift

  • Alexandru Aleman
  • William T. Ross
  • Nathan S. Feldman
Part of the Frontiers in Mathematics book series (FM)

Abstract

For a \( \mathbb{D} \) function ϑ, form the subspace
$$ K_{z\vartheta } : = H^2 \left( \mathbb{D} \right) \cap (z\vartheta H^2 (\mathbb{D}))^ \bot $$
Since z ϑ H2 \( \left( \mathbb{D} \right) \) is an S-invariant subspace of H2\( \left( \mathbb{D} \right) \), then Kzϑ will be an S*-invariant subspace of H2\( \left( \mathbb{D} \right) \), where
$$ S^* f = \frac{{f - f(0)}} {z} $$
is the backward shift operator. It is also easy to see that Kzϑ contains the constants. In fact, by Beurling’s theorem, every S*-invariant subspace, which also contains the constants. takes the form Kzϑ for some \( \mathbb{D} \)-inner function ϑ. It is well known [16, 26] that functions in Kzϑ have special ‘continuation’ properties. Indeed, recall from (3.3.2) that for hL1(m)
$$ (Ch)(\lambda ) : = \int_\mathbb{T} {\frac{{h(\zeta )}} {{\zeta - \lambda }}dm} (\zeta ) $$
denotes the Cauchy transform of h. It is known [16, p. 87] that for any fKzϑ the meromorphic function
$$ \tilde f(\lambda ) : = \frac{{C(f\overline {\zeta \vartheta } )(\lambda )}} {{C(\overline {\zeta \vartheta } )(\lambda )}} $$
(4.1.1)
on \( \mathbb{D}_e \) is a pseudocontinuation of f in that the non-tangential limits of f (from \( \mathbb{D} \) ) and \( \tilde f \) (from \( \mathbb{D}_e \)) are equal almost everywhere on \( \mathbb{T} \) . Using the Cauchy integral formula and power series, one can prove the identity
$$ \tilde f(\lambda ) = \frac{1} {{\vartheta ^* (\lambda )}}\sum\limits_{n = 1}^\infty {\frac{1} {{\lambda ^{n - 1} }}} \widehat{f\overline {\zeta \vartheta } } ( - n), $$
where \( \hat \cdot (k) \) denotes the k-th Fourier coefficient and
$$ \vartheta ^* (\lambda ): = \overline {\vartheta \left( {\begin{array}{*{20}c} 1 \\ {\overline{\overline \lambda } } \\ \end{array} } \right), } \lambda \in \mathbb{D}_e . $$
(4.1.2)
This says that
$$ \tilde f \in \frac{1} {{\vartheta ^* }}H^2 (\mathbb{D}_e ) \forall f \in K_{z\vartheta } . $$
(4.1.3)
.

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Copyright information

© Birkhäuser Verlag AG 2009

Authors and Affiliations

  • Alexandru Aleman
    • 1
  • William T. Ross
    • 2
  • Nathan S. Feldman
    • 3
  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden
  2. 2.Department of Mathematics and Computer ScienceUniversity of RichmondRichmondUSA
  3. 3.Department of MathematicsWashington & Lee UniversityLexingtonUSA

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