Computations of spectral radii on G-spaces

In a series of papers [12], [13], [14] based on ideas introduced by Soardi andWoess [17] and Salvatori [15], we have developed tools to compute the norms and/or spectral radii of Markov operators that are invariant under the left action of a group when the action is either transitive or almost transitive (i.e has a compact factor space). In [12], [13], we where concerned with countable homogeneous spaces. A typical example is given by the simple random walk on the vertex set of the (r+ 1)-regular tree T = Tr. For each vertex v, set K(v, w) = 1/(r + 1) if w is a neighbor of v, and K(v, w) = 0 otherwise. There are many groups G that act transitively on the vertex set of the tree and such that K(gv, gw) = K(v, w) for all g ∈ G. One such group is the group of those isometries of the tree that fix one end. See Figure 1 below in Section 4. This group is amenable and non-unimodular. These two properties lead to an easy computation of the norms σp(K) and spectral radii ρp(K) of the the operator K acting on Lp(T), 1 ≤ p ≤ ∞ (the reference measure here is the counting measure). In [14], the theory developed in [13] is extended to general locally compact state spaces. The purpose of the present paper is to illustrate some of the results of [14] with concrete applications and examples. In Section 2, we briefly introduce the general framework and recall the relevant results of [14], in particular the main tool, Theorem 2.1. In the short Section 3, we explain a generalization of Theorem 2.1 and its relation with an identity of Hardy, Littlewood and and Pólya [6] plus extensions of the latter. The remaining Sections are devoted to two classes of examples: Section 4 deals with diffusion operators with constant coefficients on a regular tree, seen as a 1-complex where each edge is a copy of the unit interval. Section 5 links our results with classic harmonic analysis by looking at convolution operators on semisimple and other Lie groups, as well as general locally compact groups, not always necessarily connected.