We study the problem of simultaneous stabilization for the algebra . Invertible pairs , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that 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 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 . 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 are totally reducible, that is, for which pairs there exist two units u and v in such that uf + vg = 1.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-3-4,
author = {Raymond Mortini and Brett D. Wick},
title = {Simultaneous stabilization in $A\_{$\mathbb{R}$}()$
},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {223-235},
zbl = {1197.46013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-3-4}
}
Raymond Mortini; Brett D. Wick. Simultaneous stabilization in $A_{ℝ}()$
. Studia Mathematica, Tome 192 (2009) pp. 223-235. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-3-4/