Stabilization in H∞R(D)

Brett D. Wick

doi:10.5565/PUBLMAT_54110_02

Stabilization in H^∞_R(D)

Brett D. Wick

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

It is shown that for H^∞_R(D) functions f ₁ and f₂ with inf_z∈D (|f₁(z)| + |f₂(z)|) ≥ δ > 0 and f₁ being positive on the real zeros of f₂, then there exists H^∞ _R(D) functions g₂ and g₁, g₁ ^-1 with norm controlled by a constant depending only on δ and g₁f₁ + g₂f₂ = l ∀ z ∈ D. These results are connected to the computation of the stable rank of the algebra H^∞_R(D) and to results in Control Theory.

Original language	English
Pages (from-to)	25-52
Number of pages	28
Journal	Publicacions Matematiques
Volume	54
Issue number	1
DOIs	https://doi.org/10.5565/PUBLMAT_54110_02
State	Published - 2010

Keywords

Banach algebras
Control theory
Corona theorem
Stable rank

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abstract = "It is shown that for H∞R(D) functions f 1 and f2 with infz∈D (|f1(z)| + |f2(z)|) ≥ δ > 0 and f1 being positive on the real zeros of f2, then there exists H∞ R(D) functions g2 and g1, g1 -1 with norm controlled by a constant depending only on δ and g1f1 + g2f2 = l ∀ z ∈ D. These results are connected to the computation of the stable rank of the algebra H∞R(D) and to results in Control Theory.",

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N2 - It is shown that for H∞R(D) functions f 1 and f2 with infz∈D (|f1(z)| + |f2(z)|) ≥ δ > 0 and f1 being positive on the real zeros of f2, then there exists H∞ R(D) functions g2 and g1, g1 -1 with norm controlled by a constant depending only on δ and g1f1 + g2f2 = l ∀ z ∈ D. These results are connected to the computation of the stable rank of the algebra H∞R(D) and to results in Control Theory.

AB - It is shown that for H∞R(D) functions f 1 and f2 with infz∈D (|f1(z)| + |f2(z)|) ≥ δ > 0 and f1 being positive on the real zeros of f2, then there exists H∞ R(D) functions g2 and g1, g1 -1 with norm controlled by a constant depending only on δ and g1f1 + g2f2 = l ∀ z ∈ D. These results are connected to the computation of the stable rank of the algebra H∞R(D) and to results in Control Theory.

KW - Banach algebras

KW - Control theory

KW - Corona theorem

KW - Stable rank

UR - https://www.scopus.com/pages/publications/77649320337

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DO - 10.5565/PUBLMAT_54110_02

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SN - 0214-1493

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SP - 25

EP - 52

JO - Publicacions Matematiques

JF - Publicacions Matematiques

IS - 1

ER -

Stabilization in H^∞_R(D)

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Keywords

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