On the Type of Certain Co-Semigroups

Let be the infinitesimal generator of a Co-semigroup in Hilbert space. We assume that A0 is normal and B is bounded. We further assume that there is M > 0 such that every spectral value of A:O with modulus greater than M - 1 is an isolated eigenvalue with finite multiplicity. Moreover, we assume that the multiplicities of all the eigenvalues lying in any given unit disk (centered at some z with ) do not add up to more than some fixed integer n. It is proved that the type of the semigroup is determined by the spectrum of A. Applications to one-dimensional hyperbolic problems are given.