On Γ_n-contractions and their Conditional Dilations

2017/04/14 by Pal, Avijit
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1704.04508

Abstract

We prove some estimates for elementary symmetric polynomials on \mathbb Dⁿ. We show that these estimates are sharp which allow us to study the properties of closed symmetrized polydisc Γ_n. Furthermore, we show the existence and uniqueness of solutions to the operator equations S_i-S_n-i^*S_n=D_SnX_iD_Sn~~\rmand~~S_n-i-S_i^*S_n=D_SnX_n-iD_Sn, where X_i,X_n-i∈ \mathcal B(\mathcal D_Sn), ~\rmfor ~all~ i=1,…,(n-1), with numerical radius not greater than 1, for a Γ_n-contraction (S₁,…, S_n). We construct a conditional dilation of various classes of Γ_n-contractions. Various properties of a Γ_n-contraction and its explicit dilation allow us to construct a concrete functional model for a Γ_n-contraction. We describe the structure and additional characterization of Γ_n-unitaries and Γ_n-isometries in detail.

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