Certain non-homogeneous matricial domains and Pick-Nevanlinna interpolation problem

TL;DR

This paper studies certain matricial domains linked to complex plane domains, including the spectral unit ball, and explores their automorphisms and the conditions for solving Pick-Nevanlinna interpolation problems.

Contribution

It generalizes known results about the spectral unit ball to broader matricial domains and analyzes automorphism groups and interpolation conditions.

Findings

Automorphism group of these domains is non-transitive.

Provides necessary conditions for 2-point and 3-point interpolation.

Generalizes previous results on spectral unit ball to new domains.

Abstract

In this article, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows, generalizing a result of Ransford-White for the spectral unit ball, that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider $2$-point and $3$-point Pick-Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.