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.