Simultaneous stabilization in $A_{ℝ}\left(\right)$

Raymond Mortini, and Brett D. Wick. "Simultaneous stabilization in $A_{ℝ}()$." Studia Mathematica 191.3 (2009): 223-235. <http://eudml.org/doc/284691>.

@article{RaymondMortini2009,
abstract = {We study the problem of simultaneous stabilization for the algebra $A_\{ℝ\}()$. Invertible pairs $(f_\{j\},g_\{j\})$, j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $αf_\{j\} + βg_\{j\}$ 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 $A_\{ℝ\}()$ 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 $A_\{ℝ\}()$. 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 $A_\{ℝ\}()²$ are totally reducible, that is, for which pairs there exist two units u and v in $A_\{ℝ\}()$ such that uf + vg = 1.},
author = {Raymond Mortini, Brett D. Wick},
journal = {Studia Mathematica},
keywords = {Banach algebras; control theory; corona theorem; stable rank},
language = {eng},
number = {3},
pages = {223-235},
title = {Simultaneous stabilization in $A_\{ℝ\}()$},
url = {http://eudml.org/doc/284691},
volume = {191},
year = {2009},
}

TY - JOUR
AU - Raymond Mortini
AU - Brett D. Wick
TI - Simultaneous stabilization in $A_{ℝ}()$
JO - Studia Mathematica
PY - 2009
VL - 191
IS - 3
SP - 223
EP - 235
AB - We study the problem of simultaneous stabilization for the algebra $A_{ℝ}()$. Invertible pairs $(f_{j},g_{j})$, j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $αf_{j} + βg_{j}$ 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 $A_{ℝ}()$ 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 $A_{ℝ}()$. 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 $A_{ℝ}()²$ are totally reducible, that is, for which pairs there exist two units u and v in $A_{ℝ}()$ such that uf + vg = 1.
LA - eng
KW - Banach algebras; control theory; corona theorem; stable rank
UR - http://eudml.org/doc/284691
ER -