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

Tom 191 / 2009

Raymond Mortini, Brett D. Wick Studia Mathematica 191 (2009), 223-235 MSC: Primary 46E25; Secondary 46J15, 93D99. DOI: 10.4064/sm191-3-4

Streszczenie

We study the problem of simultaneous stabilization for the algebra $A_\mathbb R(\mathbb D)$. Invertible pairs $(f_j,g_j)$, $j=1,\ldots, n$, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair $(\alpha,\beta)$ of elements such that $\alpha f_j+\beta g_j$ is invertible in this algebra for $j=1,\ldots, n$.

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 $A_\mathbb R(\mathbb D)$ 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 $A_\mathbb R(\mathbb D)$.

For $n\geq 3$ 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 $A_\mathbb R(\mathbb D)^2$ are totally reducible, that is, for which pairs there exist two units $u$ and $v$ in $A_\mathbb R(\mathbb D)$ such that $uf+vg=1$.

Autorzy

  • Raymond MortiniDépartement de Mathématiques
    LMAM, UMR 7122
    Université Paul Verlaine
    Ile du Saulcy, F-57045 Metz, France
    e-mail
  • Brett D. WickDepartment of Mathematics
    University of South Carolina
    LeConte College
    1523 Greene Street
    Columbia, SC 29208, U.S.A.
    and
    The Fields Institute
    222 College Street, 2nd Floor
    Toronto, Ontario, M5T 3J1 Canada
    e-mail

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