Universal properties of the isotropic Laplace operator on homogeneous trees

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