ON THE TYPE OF CERTAIN C(0)-SEMIGROUPS

Let A = A0 + B be the infinitesimal generator of a C0-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 A0 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 Absolute value of z less-than-or-equal-to M) 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.