Characterizations of the symmetrized polydisc via another family of domains.

We find new characterizations for the points in the symmetrized polydisc 𝔾 n , a family of domains associated with the spectral interpolation, defined by 𝔾 n : = ∑ 1 ≤ i ≤ n z i , ∑ 1 ≤ i < j ≤ n z i z j , ... , ∏ i = 1 n z i : | z i | < 1 , i = 1 , ... , n . We introduce a new family of domains which we call the extended symmetrized polydisc 𝔾 ̃ n , and define in the following way: 𝔾 ̃ n : = (y 1 , ... , y n − 1 , q) ∈ ℂ n : q ∈ 𝔻 , y j = β j + β ̄ n − j q , β j ∈ ℂ and | β j | + | β n − j | < n j , for j = 1 , ... , n − 1 . We show that 𝔾 n = 𝔾 ̃ n for n = 1 , 2 and that 𝔾 n ⊊ 𝔾 ̃ n for n ≥ 3. We first obtain a variety of characterizations for the points in 𝔾 ̃ n and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in 𝔾 n . Also, we obtain similar characterizations for the points in Γ n ∖ 𝔾 n , where Γ n = 𝔾 n ¯. A set of n − 1 fractional linear transformations plays central role in the entire program. We also show that for n ≥ 2 , 𝔾 ̃ n is nonconvex but polynomially convex and is starlike about the origin but not circled. [ABSTRACT FROM AUTHOR]