Characterizations of the symmetrized polydisc via another family of domains

Abstract

We find new characterizations for the points in the symmetrized polydisc [Formula: see text], a family of domains associated with the spectral interpolation, defined by [Formula: see text] We introduce a new family of domains which we call the extended symmetrized polydisc [Formula: see text], and define in the following way: [Formula: see text] [Formula: see text] We show that [Formula: see text] for [Formula: see text] and that [Formula: see text] for [Formula: see text]. We first obtain a variety of characterizations for the points in [Formula: see text] and we apply these necessary and sufficient conditions to produce an analogous set of characterizations for the points in [Formula: see text]. Also, we obtain similar characterizations for the points in [Formula: see text], where [Formula: see text]. A set of [Formula: see text] fractional linear transformations plays central role in the entire program. We also show that for [Formula: see text], [Formula: see text] is nonconvex but polynomially convex and is starlike about the origin but not circled.

Interpolating functions for a family of domains related to $μ$-synthesis

“…This article is a sequel of [12] and [13]. In [12] the author and Pal introduced a new family of domains, namely the extended symmetrized polydisc, G n , where G n := (y 1 , . .…”

“…But there was no Schwarz type lemma for G n , n ≥ 3. Note that G 2 = G 2 and G n G n for n ≥ 3 (see [5], [12]). In [13], the author and Pal produced a Schwarz lemma for G n and G n and showed that an interpolating function, when exists, may not be unique.…”

“…Note that G 2 = G 2 and G n G n for n ≥ 3 (see [5], [12]). In [13], the author and Pal produced a Schwarz lemma for G n and G n and showed that an interpolating function, when exists, may not be unique. The aim of this paper is to describe a class of analytic interpolants when we have such a Schwarz lemma for G n .…”

“…The closure of G n is denoted by Γ n . to study complex geometry of G n and Γ n , in [12], we introduced (n − 1) fractional linear transformations Φ 1 , . .…”

“…Theorems 2.5 and 2.7 of [12] characterize the points in G n and Γ n . The extended symmetrized polydisc is a non-convex but polynomially convex domain for all n. Also G n is a starlike domain (see [12]). Let B 1 , .…”