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 n of the kernels of (Δ − γI) . 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.
Universal properties of the isotropic Laplace operator on homogeneous trees
Joel M. Cohen;Mauro Pagliacci;
2022
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 n of the kernels of (Δ − γI) . 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.File in questo prodotto:
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