  We study the operators T on the weighted space L^p commuting either with the
right translations St or left translations P^+S_{-t} and we establish the
existence of a symbol of T. We characterize completely the spectrum of St. We
obtain a similar result for the spectrum of P^+S_{-t} and some spectral results
for the bounded operators commuting with (St), t>0 or with (P^+St), t<0.
