Nearly invariant and the backward shift
Alexandru Aleman, William T. Ross, Nathan S. Feldman · Frontiers in mathematics · 2009
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 h∈L1(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 f∈Kzϑ the meromorphic function 4.1.1 $$ \tilde f(\lambda ) : = \frac{{C(f\overline {\zeta \vartheta } )(\lambda )}} {{C(\overline {\zeta \vartheta } )(\lambda )}} $$ 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 4.1.2 $$ \vartheta ^* (\lambda ): = \overline {\vartheta \left( {\begin{array}{*{20}c} 1 \\ {\overline{\overline \lambda } } \\ \end{array} } \right), } \lambda \in \mathbb{D}_e . $$ This says that 4.1.3 $$ \tilde f \in \frac{1} {{\vartheta ^* }}H^2 (\mathbb{D}_e ) \forall f \in K_{z\vartheta } . $$ .