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Corona conditions and symbols with a gap around zero

Convolution equations on a finite interval (which we can assume to
be $[0,1]$) lead to the problem of factorizing matrix functions
$$G=\left[
\begin{array}{cc}
  e_{-1} & 0 \\
  g & e_1 \\
\end{array}
\right]$$ where $e_\theta(\xi)=e^{i\theta\xi}$,
$\theta\in\mathbb{R}$ and $g\in L_\infty (\mathbb{R})$.

Here we consider $g$ of the form
$$g=a_+e_\mu + a_-e_{-\sigma}$$
with $a_\pm\in H_\infty (\mathbb{C}^\pm)$ and $\mu, \sigma >0$.
Imposing some corona-type conditions on $a_\pm$, we show that
solutions to the Riemann-Hilbert problem $Gh_+=h_-$, with
$h_\pm\in (H_\infty (\mathbb{C}^\pm))^2$, can be determined
explicitly and conditions for invertibility of the Toeplitz
operator with symbol $G$ in $(H_p^+)^2$ can be derived from them.
 
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