We study in this paper the spectrum of some kernels acting on a locally finite tree, in particular those associated to an isotropic random walk on the tree with jumps of length 0, 1 or 2. Such a kernel is a function R on S_×_S where S is the set of vertices of the tree, it acts on lr(S). We always assume the kernel R to be invariant under the action of a group Λ of authomorphisms almost transitive on S. This work generalizes results of A. Figa Talamanca and T. Steger who deal with homogeneous trees and a fixed group Λ, simply transitive on S; it shows the diversity of the spectrum depending on the invariance group.
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Ferdaous Kellil, Guy Rousseau, Etude du Spectre Pour Certains Noyaux sur un Arbre. Rend. Sem. Mat. Univ. Padova 120 (2008), pp. 29–44DOI 10.4171/RSMUP/120-2