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 ℓ2-spectrum spec(P), i.e., the eigenfunctions with eigenvalue γ=λ−1 of the Laplace operator Δ=P−I, and also the λ-polyharmonic functions, that is, the union of the kernels of (Δ−γI)n. We prove that, on a suitable Banach space generated by the λ-polyharmonic functions, the operator eΔ−γI is hypercyclic, although Δ−γI is not.