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Simultaneous stabilization in $A_{ℝ} ()$

Raymond Mortini; Brett D. Wick — 2009

Studia Mathematica

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 $α f_{j} + β g_{j}$ 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....

A remark on the multipliers on spaces of Weak Products of functions

Stefan Richter; Brett D. Wick — 2016

Concrete Operators

If H denotes a Hilbert space of analytic functions on a region Ω ⊆ Cd , then the weak product is defined by [...] We prove that if H is a first order holomorphic Besov Hilbert space on the unit ball of Cd , then the multiplier algebras of H and of H ⊙ H coincide.

Doubly commuting submodules of the Hardy module over polydiscs

Jaydeb Sarkar; Amol Sasane; Brett D. Wick — 2013

Studia Mathematica

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 $H^{\infty} (ⁿ)$ .