Characterizations of symmetrized polydisc

Abstract

Let Γn, n ≥ 2, denote the symmetrized polydisc in ℂn, and Γ1 be the closed unit disc in ℂ. We provide some characterizations of elements in Γn. In particular, an element (s1,..., sn−1, p) ∈ ℂn is in Γn if and only if \({s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p\), j = 1,..., n − 1, for some (β1,..., βn−1) ∈ Γn−1, and |p| ≤ 1.