Function theoretic aspects of the symmetrized polydisc and generalization

We introduce Schur-Agler type class for the symmetrized polydisc $\mathbb{G}_d$ and establish a realization theorem for functions in this class. We state and prove an interpolation theorem on $\mathbb{G}_d$ with interpolating functions belonging to the associated Schur-Agler type class. Moreover, Toeplitz corona and extension theorems are established for $\mathbb{G}_d$. We also extend the realization, interpolation, Toeplitz corona and extension theorems to a more general symmetrized family of domains $\Theta_d$.