On interpolation in Carathéodory hyperbolic domains

We study the relation between Pick bodies on Carathéodory hyperbolic domains and contractions on finite dimensional Hilbert spaces. We give a condition sufficient to realize Pick bodies on Carathéodory hyperbolic domains as a Pick body on the open unit disc.

💡 Research Summary

The paper investigates Pick interpolation bodies on Carathéodory hyperbolic domains, extending the classical Pick theory from the unit disc to much more general complex domains. For a domain Ω⊂ℂ^m that is Carathéodory hyperbolic (i.e., the Carathéodory pseudo‑distance is non‑degenerate), the authors define the Pick body

 D_Ω(z₁,…,zₙ)= { (f(z₁),…,f(zₙ)) : f∈O(Ω,𝔻) },

where O(Ω,𝔻) denotes all holomorphic maps from Ω into the closed unit disc 𝔻. In the classical case Ω=𝔻, the Pick problem is solved by the positivity of the matrix