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

Brett D. Wick

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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.

Article information

Publ. Mat., Volume 54, Number 1 (2010), 25-52.

First available in Project Euclid: 8 January 2010

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E25: Rings and algebras of continuous, differentiable or analytic functions {For Banach function algebras, see 46J10, 46J15} 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]

Banach Algebras control theory Corona theorem stable rank


Wick, Brett D. Stabilization in $H^\infty_{\mathbb{R}}(\mathbb{D})$. Publ. Mat. 54 (2010), no. 1, 25--52.

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See also

  • See also: Brett D. Wick. Corrigendum: "Stabilization in \boldmath$H^\infty_{\mathbb{R}}(\mathbb{D})$". Publ. Mat. Volume 55, Number 1 (2011), 251-260.