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Universal properties of the isotropic Laplace operator on homogeneous trees (2202.07772v2)

Published 15 Feb 2022 in math.FA

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 $\ell2$ 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 $n\geqslant 0$. 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.

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