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

The last-named author acknowledges support by MIUR Excellence Departments Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006, and by Istituto Nazionale di Alta Matematica, Gruppo GNAFA. Adv. Math. (2022)

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• Universal properties of the isotropic Laplace operator on homogeneous trees

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• Universal properties of the isotropic Laplace operator on homogeneous trees