Universal properties of the isotropic Laplace operator on homogeneous trees

Abstract

Let $P$ be the isotropic nearest neighbor transition operator on a homogeneous tree. We consider the $λ$-eigenfunctions of $P$ for $λ$ outside its $\ell^2$ spectrum, i.e., the eigenfunctions with eigenvalue $γ=λ- 1$ of the Laplace operator $Delta=P- \mathbb I$, and also the $λ-$polyharmonic functions, that is, the union of the kernels of $(Delta-γ\mathbb I)^n$ for $n\geqslant 0$. We prove that, on a suitable Banach space generated by the $λ-$polyharmonic functions, the operator $e^{Delta-γ\mathbb I}$ is hypercyclic, although $Delta-γ\mathbb I$ is not.