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Let T be a semigroup of linear contractions on a Banach space X, and let $X_s\left(T\right)=x\in X:li{m}_{s\to \infty }\parallel T\left(s\right)x\parallel =0$. Then $X_s\left(T\right)$ is the annihilator of the bounded trajectories of T*. If the unitary spectrum of T is countable, then $X_s\left(T\right)$ is the annihilator of the unitary eigenvectors of T*, and $li{m}_s\parallel T\left(s\right)x\parallel =inf\parallel x-y\parallel :y\in X_s\left(T\right)$ for each x in X.