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. |