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

TL;DR

This paper investigates the conditions for simultaneous stabilization in the algebra $A_ ( ext{D})$, revealing that for two pairs, specific criteria enable stabilization, but for three or more pairs, stabilization is generally impossible.

Contribution

It provides necessary and sufficient conditions for simultaneous stabilization in $A_ ( ext{D})$ when $n=2$, and demonstrates the impossibility for $n extgreater 2$, linking to total reducibility of pairs.

Findings

For $n=2$, explicit conditions for stabilization are established.

For $n extgreater 2$, simultaneous stabilization is generally not possible.

The work connects stabilization to the concept of total reducibility of pairs.

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