Distinguished varieties in the polydisc and dilation of commuting contractions

2022/05/01 by Pal, Sourav · 1 citation
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2205.00540

Abstract

A distinguished variety in the polydisc \mathbb Dⁿ is an affine complex algebraic variety that intersects \mathbb Dⁿ and exits the domain through the n-torus \mathbb Tⁿ without intersecting any other part of the topological boundary of \mathbb Dⁿ. We find two different characterizations for a distinguished variety in the polydisc \mathbb Dⁿ in terms of the Taylor joint spectrum of certain linear matrix-pencils and thus generalize the seminal work due to Agler and M\raise.45ex\hboxcCarthy [Acta Math., 2005] on distinguished varieties in \mathbb D². We show that a distinguished variety in \mathbb Dⁿ is a part of an affine algebraic curve which is a set-theoretic complete intersection. We also show that if (T₁, … , T_n) is commuting tuple of Hilbert space contractions such that the defect space of T=∏_i=1ⁿ T_i is finite dimensional, then (T₁, … , T_n) admits a commuting unitary dilation (U₁, … , U_n) with U=∏_i=1ⁿ U_i being the minimal unitary dilation of T if and only if some certain matrices associated with (T₁, … , T_n) define a distinguished variety in \mathbb Dⁿ.

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• Constrained dilation and Γ-contractions

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