Universal Jamison spaces and Jamison sequences for $C_0$-semigroups
Volume 214 / 2013
Abstract
An increasing sequence $(n_k)_{k\ge 0}$ of positive integers is said to be a Jamison sequence if for every separable complex Banach space $X$ and every $T\in \mathcal {B}(X)$ which is partially power-bounded with respect to $(n_k)_{k\ge 0}$, the set $\sigma _p(T)\cap \mathbb {T}$ is at most countable. We prove that for every separable infinite-dimensional complex Banach space $X$ which admits an unconditional Schauder decomposition, and for any sequence $(n_k)_{k\ge 0}$ which is not a Jamison sequence, there exists $T\in \mathcal {B}(X)$ which is partially power-bounded with respect to $(n_k)_{k\ge 0}$ and has the set $\sigma _p(T)\cap \mathbb {T}$ uncountable. We also investigate the notion of Jamison sequences for $C_0$-semigroups and we give an arithmetic characterization of such sequences.
