Exponential convergence of Markovian semigroups and their spectra on L ^p-spaces

Markovian semigroups on L²-space with suitable conditions can be regarded as Markovian semigroups on L^p-spaces for p ∈ [1,∞). When we additionally assume the ergodicity of the Markovian semigroups, the rate of convergence on L^p-space for each p is considerable. However, the rate of convergence depends on the norm of the space. The purpose of this paper is to investigate the relation between the rates on L^p-spaces for different p,s, to obtain some sufficient condition for the rates to be independent of p, and to give an example for which the rates depend on p.We also consider spectra of Markovian semigroups on L ^p-spaces, because the rate of convergence is closely related to the spectra.