Etude du Spectre Pour Certains Noyaux sur un Arbre

We study in this paper the spectrum of some kernels acting on a locally finite tree, in particular those associated to an anisotropic random walk on a tree, with jumps of length 0, 1 or 2. Such a kernel is a function R on SxS where S is the set of vertices of the tree, it acts on l^r(S). We always assume the kernel R to be invariant under the action of a group G of automorphisms almost transitive on S. This work generalizes results of A. Figa Talamanca and T. Steger who deal with homogeneous trees and a fixed group G, simply transitive on S; it shows the diversity of the spectrum depending on the invariance group.