Characterizations of the symmetrized polydisc via another family of domains

Abstract

We find new characterizations for the points in the \textit{symmetrized polydisc} $\mathbb G_n$, a family of domains associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots, \prod_{i=1}^n z_i \right): \,|z_i|<1, i=1,\dots,n \right \}. \] We introduce a new family of domains which we call \textit{the extended symmetrized polydisc} $\widetilde{\mathbb G}_n$, and define in the following way: \begin{align*} \widetilde{\mathbb G}_n := \Bigg\{ (y_1,\dots,y_{n-1}, q)\in \mathbb C^n :\; q \in \mathbb D, \; y_j = \beta_j + \bar{\beta}_{n-j} q, \; \beta_j \in \mathbb C &\text{ and }\\ |\beta_j|+ |\beta_{n-j}| < {n \choose j} &\text{ for } j=1,\dots, n-1 \Bigg\}. \end{align*} We show that $\mathbb G_n=\widetilde{\mathbb G}_n$ for $n=1,2$ and that ${\mathbb G}_n \subsetneq \widetilde{\mathbb G}_n$ for $n\geq 3$. We first obtain a variety of characterizations for the points in $\widetilde{\mathbb G}_n$ and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in ${\mathbb G}_n$. Also we obtain similar characterizations for the points in $\Gamma_n \setminus {\mathbb G}_n$, where $\Gamma_n =\overline{{\mathbb G}_n}$. A set of $n-1$ fractional linear transformations play central role in the entire program. We also show that for $n\geq 2$, $\widetilde{\mathbb G}_n$ is non-convex but polynomially convex and is starlike about the origin but not circled.