We study the spectrum Sp(T) of a bounded, strongly continuous, representation T of a suitable Abelian semigroup S on a Banach space X. In particular, we consider the case when the unitary part of Sp(T) is countable (or scattered) and the unitary part of the point spectrum of \(T^*\) is empty. It is known from previous work of the authors (in collaboration with W. Arendt and Yu. I. Lyubich) that \(\inf_{t\in S}\| T(t)x\| =0\) for each x in X, if \(S={\mathbb{R}}_+\) or if T is norm-continuous. We show that this result remains true without these constraints. In passing, we establish that a finite number of commuting \(C_ 0\)-semigroups of isometries have a common sequence of approximate eigenvectors.