On ${\mathrm{\Gamma }}_n$-contractions and their Conditional Dilations

We prove some estimates for elementary symmetric polynomials on $\mathbb D^n.$ We show that these estimates are sharp which allow us to study the properties of closed symmetrized polydisc ${\mathrm{\Gamma }}_n\mathrm{.}$ Furthermore, we show the existence and uniqueness of solutions to the operator equations $$S_i-S_{n-i}^{\ast }{S}_n=D_{S_n}{X}_i{D}_{S_n}\mathrm{a}\mathrm{n}\mathrm{d}{S}_{n-i}-S_i^{\ast }{S}_n=D_{S_n}{X}_{n-i}{D}_{S_n},$$ where $X_i,X_{n-i}\in \mathcal B(\mathcal D_{S_n}), ~{\rm{for ~all~}} i=1,\ldots,(n-1),$withnumericalradiusnotgreaterthan$1,$fora$\Gamma_n$-contraction$(S_1,\ldots, S_n).$ We construct a conditional dilation of various classes of ${\mathrm{\Gamma }}_n$-contractions. Various properties of a ${\mathrm{\Gamma }}_n$-contraction and its explicit dilation allow us to construct a concrete functional model for a ${\mathrm{\Gamma }}_n$-contraction. We describe the structure and additional characterization of ${\mathrm{\Gamma }}_n$-unitaries and ${\mathrm{\Gamma }}_n$-isometries in detail.