Size of the peripheral point spectrum under power or resolvent growth conditions

We characterize Jamison sequences, that is sequences (nk) of positive integers with the following property: every bounded linear operator T acting on a separable Banach space with supk T nk < +∞ has a countable set of peripheral eigenvalues. We also discuss partially power-bounded operators acting on Banach or Hilbert spaces having peripheral point spectra with large Hausdorff dimension. For a Lavrentiev domain Ω in the complex plane, we show the uniform minimality of some families of eigenvectors associated with peripheral eigenvalues of operators satisfying the Kreiss resolvent condition with respect to Ω. We introduce and study the notion of Ω-Jamison sequence, which is defined by replacing the partial power-boundedness condition supk T nk < +∞ by supk FΩ nk (T ) < +∞, where FΩ n is the nth Faber polynomial of Ω. A characterization of Ω-Jamison sequences is obtained for domains with sufficiently smooth boundary. © 2007 Elsevier Inc. All rights reserved