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

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)})=\delta>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)}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(\delta,\epsilon) and f_1(z)g_1(z)+f_2(z)g_2(z)=1\quad\forall z\in\D. Comment: 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 sam