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An increasing sequence ${\left(n_k\right)}_{k\ge 0}$ of positive integers is said to be a Jamison sequence if for every separable complex Banach space X and every T ∈ ℬ(X) which is partially power-bounded with respect to ${\left(n_k\right)}_{k\ge 0}$, the set ${\sigma }_p\left(T\right)\cap$ 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 ${\left(n_k\right)}_{k\ge 0}$ which is not a Jamison sequence, there exists T ∈ ℬ(X) which is partially power-bounded with respect to ${\left(n_k\right)}_{k\ge 0}$ and has the...