Point spectra of partially power-bounded operators

Let T be an operator on a separable Banach space, and denote by p(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 p(T ) ∩ T be uncountable, yet T n ∞. We further investigate the relationship between the growth of sequences (nk) such that supk T nk < ∞ and the possible size of p(T ) ∩ T. Analogous results are also derived for continuous operator semigroups (Tt )t 0. © 2005 Elsevier Inc. All rights reserved