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

• Autorzy: Raymond Mortini , Brett D. Wick
• Języki publikacji: EN

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.

Kategorie tematyczne

• 93D99: None of the above, but in this section
• 46E25: Rings and algebras of continuous, differentiable or analytic functions
• 46J15: Banach algebras of differentiable or analytic functions, H p -spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2009
• Tom: 191
• Numer: 3
• Strony: 223-235

Daty

wydano

2009

Twórcy

autor

Raymond Mortini

• Département de Mathématiques, LMAM, UMR 7122, Université Paul Verlaine, Ile du Saulcy, F-57045 Metz, France

autor

Brett D. Wick

• 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

Identyfikatory

DOI

10.4064/sm191-3-4