Triangular Tetrablock-contractions, factorization of contractions, dilation and subvarieties

2022/04/25 by Pal, Sourav
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2204.11387

Abstract

A commuting triple of Hilbert space operators (A,B,P), for which the closed tetrablock \mathbb E is a spectral set, is called a tetrablock-contraction or simply an \mathbb E-contraction, where \mathbb E=\(a₁₁,a₂₂, det A): A=[a_ij]∈ \mathcal M₂(\mathbb C), ‖A‖ lt;1 \ ⊂ \mathbb C³ is a polynomially convex domain which is naturally associated with the μ-synthesis problem. We introduce triangular \mathbb E-contractions and prove that every pure triangular \mathbb E-contraction dilates to a pure triangular \mathbb E-isometry. We construct a functional model for a pure triangular \mathbb E-isometry and apply that model to find a new proof for the famous Berger-Coburn-Lebow Model Theorem for commuting isometries. Next we give an alternative proof to the more generalized version of Berger-Coburn-Lebow Model, namely the factorization of a pure contraction due to Das, Sarkar and Sarkar (Adv. Math. 322 (2017), 186 -- 200). We find a necessary and sufficient condition for the existence of \mathbb E-unitary dilation of an \mathbb E-contraction (A,B,P) on the smallest dilation space and show that it is equivalent to the existence of a distinguished variety in \mathbb E when the defect space D_P^* is finite dimensional.

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• Rational dilation of tetrablock contractions revisited
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