Matrix representation of distinguished varieties on symmetrised bidisc

Abstract

The symmetrisation map Π(𝑧,𝑤) → (𝑧+𝑤,𝑧𝑤) is the map given by Π(𝑧,𝑤) → (𝑧+𝑤,𝑧𝑤) and the symmetrised bidisc, 𝔾 is given by Π(𝔻2), where 𝔻 is the open unit disc in ℂ. A non empty set W ⊆ ℂ2 is called a Distinguished Variety on 𝔾 if there exists a polynomial 𝑞(𝑠,𝑝) ∈ ℂ[𝑠,𝑝] such that 𝑊 = {(𝑠,𝑝) ∈ 𝔾:𝑞(𝑠,𝑝) = 0} and Z(𝑞) ⊆ 𝔾∪𝑏𝕋∪Ω where 𝑏Γ is the image of 𝕋 under Π and Ω is the image of 𝔼 under Π. It was proven that, any Distinguished Variety on 𝔾 can be represented via a square matrix with numerical radius (𝜔(𝐴)) less than 1. Conversely, given a Distinguished Variety on 𝔾 there exists a square matrix 𝐴 with 𝜔(𝐴) ≤ 1 such that 𝑊 = {(𝑠,𝑝) ∈ 𝔾:det(𝐴 + 𝑝𝐴∗ − 𝑠𝐼) = 0}. In this work, we prove that if 𝐴𝑘 is a square matrix representing the Distinguished Variety 𝑊𝑘 on 𝔾 for 𝑘 = 1,….,𝑛 then, ⨁ 𝑛 𝑘=1 𝐴𝑘 represents the Distinguished Variety 𝑊 = ⋃ 𝑛 𝑘=1 𝑊𝑘 on 𝔾. This result would allow us to generate examples for matrices representing Distinguished Varieties on 𝔾, specially for reducible Distinguished Varieties, by taking direct sums of the matrices representing its irreducible components.