Stabilization in $H_{\mathbb{R}}^{\infty }(\mathbb{D})$

Abstract

In this paper we prove the following theorem: Suppose that $f_1,f_2\in H^\infty_\R(\D)$, with $\norm{f_1}_\infty,\norm{f_2}_{\infty}\leq 1$, with $$ \inf_{z\in\D}(\abs{f_1(z)}+\abs{f_2(z)})=δ>0. $$ Assume for some $\epsilon >0$ and small, $f_1$ is positive on the set of $x\in (-1,1)$ where $\abs{f_2(x)}<ε$ for some $\epsilon >0$ sufficiently small. Then there exists $g_1, g_1^{-1}, g_2\in H^\infty_\R(\D)$ with $$ \norm{g_1}_\infty,\norm{g_2}_\infty,\norm{g_1^{-1}}_\infty\leq C(δ,ε) $$ and $$ f_1(z)g_1(z)+f_2(z)g_2(z)=1\quad\forall z\in\D. $$

v1: 22 pages, 2 figures, to appear in Pub. Mat; v2: 32 pages, 5 figures. The earlier version incorrectly claimed a characterization, as was pointed out by R. Mortini. A key hypothesis was strengthened with the main result remaining the same