Transition operators on co-compact G-spaces.

Saloff-Coste, Laurent, and Woess, Wolfgang. "Transition operators on co-compact G-spaces.." Revista Matemática Iberoamericana 22.3 (2006): 747-799. <http://eudml.org/doc/41991>.

@article{Saloff2006,
abstract = {We develop methods for studying transition operators on metric spaces that are invariant under a co-compact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce reduced transition operators on the compact factor space whose norms and spectral radii are upper bounds for the Lp-norms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and reduced operators coincide, and under additional hypotheses, this is also sufficient for amenability. Further bounds involve the modular function of the group.In this framework, we prove among other things that the bottom of the spectrum of the Laplacian on a co-compact Riemannian manifold is 0 if and only if the group is amenable and unimodular. The same result holds for Euclidean simplicial complexes. On a geodesic, proper metric space with co-compact isometry group action, the averaging operator over balls with a fixed radius has norm equal to 1 if and only if the group is amenable and unimodular. The technique also allows explicit computation of spectral radii when the group is amenable.},
author = {Saloff-Coste, Laurent, Woess, Wolfgang},
journal = {Revista Matemática Iberoamericana},
keywords = {Cadenas de Markov; Probabilidad de transición; Operador laplaciano; G-espacios; Invariancia; -space; proper action; isometries; invariant transition operator; amenability},
language = {eng},
number = {3},
pages = {747-799},
title = {Transition operators on co-compact G-spaces.},
url = {http://eudml.org/doc/41991},
volume = {22},
year = {2006},
}

TY - JOUR
AU - Saloff-Coste, Laurent
AU - Woess, Wolfgang
TI - Transition operators on co-compact G-spaces.
JO - Revista Matemática Iberoamericana
PY - 2006
VL - 22
IS - 3
SP - 747
EP - 799
AB - We develop methods for studying transition operators on metric spaces that are invariant under a co-compact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce reduced transition operators on the compact factor space whose norms and spectral radii are upper bounds for the Lp-norms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and reduced operators coincide, and under additional hypotheses, this is also sufficient for amenability. Further bounds involve the modular function of the group.In this framework, we prove among other things that the bottom of the spectrum of the Laplacian on a co-compact Riemannian manifold is 0 if and only if the group is amenable and unimodular. The same result holds for Euclidean simplicial complexes. On a geodesic, proper metric space with co-compact isometry group action, the averaging operator over balls with a fixed radius has norm equal to 1 if and only if the group is amenable and unimodular. The technique also allows explicit computation of spectral radii when the group is amenable.
LA - eng
KW - Cadenas de Markov; Probabilidad de transición; Operador laplaciano; G-espacios; Invariancia; -space; proper action; isometries; invariant transition operator; amenability
UR - http://eudml.org/doc/41991
ER -