Stabilization in $H^\infty_{\mathbb{R}}(\mathbb{D})$

Abstract

It is shown that for $H^\infty_\mathbb{R}(\mathbb{D})$ functions $f_1$ and $f_2$ with

$\inf_{z\in\mathbb{D}}(\vert f_1(z)\vert+\vert f_2(z)\vert)\geq\delta>0$

and $f_1$ being positive on the real zeros of $f_2$, then there exists $H^\infty_\mathbb{R}(\mathbb{D})$ functions $g_2$ and $g_1$, $g_1^{-1}$ with norm controlled by a constant depending only on $\delta$ and

$g_1f_1+g_2f_2=1\quad\forall\; z\in\mathbb{D}$.

These results are connected to the computation of the stable rank of the algebra $H^\infty_\mathbb{R}(\mathbb{D})$ and to results in Control Theory.