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## Studia Mathematica

2009 | 191 | 3 | 223-235
Tytuł artykułu

### Simultaneous stabilization in $A_{ℝ}(𝔻)$

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Abstrakty
EN
We study the problem of simultaneous stabilization for the algebra $A_{ℝ}(𝔻)$. Invertible pairs $(f_{j},g_{j})$, j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $αf_{j} + βg_{j}$ is invertible in this algebra for j = 1,..., n.
For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ}(𝔻)$ has stable rank two, we are faced here with a different situation. When n = 2, necessary and sufficient conditions are given so that we have simultaneous stability in $A_{ℝ}(𝔻)$.
For n ≥ 3 we show that under these conditions simultaneous stabilization is not possible and further connect this result to the question of which pairs (f,g) in $A_{ℝ}(𝔻)²$ are totally reducible, that is, for which pairs there exist two units u and v in $A_{ℝ}(𝔻)$ such that uf + vg = 1.
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Tom
Numer
Strony
223-235
Opis fizyczny
Daty
wydano
2009
Twórcy
autor
• Département de Mathématiques, LMAM, UMR 7122, Université Paul Verlaine, Ile du Saulcy, F-57045 Metz, France
autor
• Department of Mathematics, University of South Carolina, LeConte College, 1523 Greene Street, Columbia, SC 29208, U.S.A.
• The Fields Institute, 222 College Street, 2nd Floor, Toronto, Ontario, M5T 3J1 Canada
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