Simultaneous stabilization in
Volume 191 / 2009
Abstract
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.