Relatively Bounded Rank One Perturbations of Non-Analytic Semigroups Can Generate Large Point Spectrum

It is known that relatively bounded compact perturbations of analytic semigroups yield again generators of analytic semigroups. On the other hand, for each non-analytic semigroup there is a relatively bounded rank one perturbation which is no longer a generator. In this note we show that in fact any non-analytic semigroup admits a relatively bounded rank one perturbation with a sequence of eigenvalues z_j such that the real parts R(z_j) converge to infinity.