Interpolating sequences for the Banach algebras generated by a class of test functions

Abstract

Given a domain $\Omega$ in $\mathbb{C}^n$ and a collection of test functions $\Psi$ on $\Omega$, we consider the complex-valued $\Psi$-Schur-Agler class associated to the pair $(\Omega,\,\Psi)$. In this article, we characterize interpolating sequences for the associated Banach algebra of which the $\Psi$-Schur-Agler class is the closed unit ball. When $\Omega$ is the unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$ and the class of test function includes only the identity function on $\mathbb{D}$, the aforementioned algebra is the algebra of bounded holomorphic functions on $\mathbb{D}$ and in this case, our characterization reduces to the well known result by Carleson. Furthermore, we present several other cases of the pair $(\Omega,\,\Psi)$, where our main result could be applied to characterize interpolating sequences which also show the efficacy of our main result.