A SCATTERING-THEORY ON L1 (MU) SPACES

Let {S (t); t is-an-element-of R} he a positive and bounded (c0) group on a L1 (mu) space with generator T. Let B is-an-element-of L (D (T); X) be a positive operator. We prove that the generation of a bounded group {V (t); t is-an-element-of R} by T + B as well as the existence of the wave operators s lim t --> =/- infinity V (t)S (-t) and s lim t --> + infinity S(-t) V (t) are intimately connected to the existence of the stong limits s lim lambda --> 0 +/- B (lambda-T)-1 and to their spectral radii. The results are optimal.