Point spectra of partially power-bounded operators

Let T be an operator on a separable Banach space, and denote by σp(T) its point spectrum. We answer a question left open in (Israel J. Math. 146 (2005) 93–110) by showing that it is possible that σp(T)∩T be uncountable, yet ∥Tn∥↛∞. We further investigate the relationship between the growth of sequences (nk) such that supk∥Tnk∥<∞ and the possible size of σp(T)∩T.Analogous results are also derived for continuous operator semigroups (Tt)t⩾0.