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Simultaneous stabilization in $A_{ℝ}(𝔻)$

100%

Mortini R., Wick B.

Studia Mathematica

2009

tom 191

nr 3

223-235

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 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 $A_{ℝ}(𝔻)$ 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 ≥ 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 there exist two units u and v in $A_{ℝ}(𝔻)$ such that uf + vg = 1.

Doubly commuting submodules of the Hardy module over polydiscs

81%

Sarkar J., Sasane A., Wick B.

Studia Mathematica

2013

tom 217

nr 2

179-192

In this note we establish a vector-valued version of Beurling's theorem (the Lax-Halmos theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the "weak" completion problem in $H^{∞}(𝔻ⁿ)$.

Ograniczanie wyników

2 Studia Mathematica

2 Wick B.

1 Mortini R.

1 Sarkar J.

1 Sasane A.

1 2013

1 2009

Wyniki wyszukiwania

Simultaneous stabilization in $A_{ℝ}(𝔻)$

Doubly commuting submodules of the Hardy module over polydiscs