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Let T be a representation of a suitable abelian semigroup S by isometries on a Banach space. We study the spectral conditions which will imply that T(s) is invertible for each s in S. On the way we analyse the relationship between the spectrum of T, Sp(T,S), and its unitary spectrum $S{p}_u(T,S)$. For $S={\mathbb{Z}}_{+}^n$ or ${ℝ}_{+}^n$, we establish connections with polynomial convexity.