# 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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© 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