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

Abstract

In this note 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 \textit{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 $j=1,...,n$. 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 $n\ge 3$ 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 do there exist two units $u$ and $v$ in $A_\R(\D)$ such that $uf+vg=1$.

12 pages, to appear in Studia Math

Cited by in corpus (1)

• A short proof of Cartan's Nullstellensatz for entire functions in ${\mathbb{C}}^n$