TRANSITION OPERATORS, GROUPS, NORMS, AND SPECTRAL RADII

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 stab ilizers. We give upper bounds for the norm and spectral radius of P ac ting on l(s)(X, mu), where I < s < infinity and mu is a measure on X s atisfying a compatibility condition with respect to G. When G is amena ble, 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 redu ction to a finite matrix. For unimodular groups there are further simp lifications. A variety of examples is given, including the (linear) bu ildings of type <(A)over tilde(n-1)> associated with PGL(n,F) over a l ocal field F. These results extend previous work of Soardi and Woess, Salvatori, and Saloff-Coste and Woess, where only reversible Markov op erators and the case s = 2 were studied.