An analog of the Schwartz theorem on spectral analysis on a hyperbolic plane

Abstract

Let \( \mathbb{D} \) be an open unit disk in the complex plane. It is shown that every subspace in C(\( \mathbb{D} \)) invariant under weighted conformal shifts contains a radial eigenfunction of the corresponding invariant differential operator. This function can be expressed via the Gauss hypergeometric function and is a generalization of the spherical function on the disk \( \mathbb{D} \) which is considered as a hyperbolic plane with the corresponding Riemannian structure.