Simultaneous Stabilization in Ar(d)
نویسنده
چکیده
In this note we study the problem of simultaneous stabilization for the algebra AR(D). Invertible pairs (fj , gj), j = 1, . . . , n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α, β) of elements such that αfj + βgj 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 AR(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 AR(D). 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 AR(D) 2 are totally reducible; that is, for which pairs do there exist two units u and v in AR(D) such that uf + vg = 1.
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