On Γn-contractions and their conditional dilations

Abstract

We prove some estimates for elementary symmetric polynomials on Dn. We show that these estimates are sharp which allow us to study the properties of closed symmetrized polydisc Γn. Furthermore, we show the existence and uniqueness of solutions to the operator equationsSi−Sn−i⁎Sn=DSnXiDSnandSn−i−Si⁎Sn=DSnXn−iDSn, where Xi,Xn−i∈B(DSn), with numerical radius ω(Xi+zXn−i)≤(n−1i)+(n−1n−i) for all i=1,…,(n−1) and z∈T, for a Γn-contraction (S1,…,Sn). We construct a conditional dilation of various classes of Γn-contractions. Various properties of a Γn-contraction and its explicit dilation allow us to construct a concrete functional model for a Γn-contraction. We describe the structure and additional characterization of Γn-unitaries and Γn-isometries in detail.