Let P be a transition
operator over a countable set which is invariant under the action of a locally
compact group G with compact point stabilizers. We give upper bounds for the
norm and spectral radius of P acting on ℓs(X,μ), where 1 < s < ∞ and
μ is a measure on X satisfying a compatibility condition with respect to
G. When G is amenable, our inequalities become equalities involving the
modular function of G. When G, besides being amenable, acts with finitely
many orbits then this allows easy computation of norms and spectral radii
via reduction to a finite matrix. For unimodular groups there are further
simplifications. A variety of examples is given, including the (linear) buildings of
type Ãn−1 associated with PGL(n,F) over a local field F. These results
extend previous work of Soardi and Woess, Salvatori, and Saloff-Coste and
Woess, where only reversible Markov operators and the case s = 2 were
studied.
