Bounded analytic functions on certain symmetrized domains

Mainak Bhowmik, Poornendu Kumar

Published 2022-08-16Version 1

In this note, we first give a canonical structure of scalar-valued rational inner functions on the symmetrized polydisc. Then we generalize Caratheodory's approximation result to the setting of matrix-valued holomorphic functions on the symmetrized bidisc, $\mathbb{G}$. We approximate matrix-valued holomorphic functions with sup-norm not greater than one by rational iso-inner or coiso-inner functions uniformly on compact subsets of the symmetrized bidisc using the interpolation results. En route, we also prove that any solvable data with initial nodes on $\mathbb{G}$ and the final nodes in the operator norm unit ball of the rectangular matrices has a rational iso-inner or coiso-inner solution. Finally, we give necessary and sufficient conditions for matrix-valued Schur class functions on $\mathbb{G}$ to have a Schur class factorization on $\mathbb{G}$ in several situations.