Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem $u^{\left(n\right)}\left(t\right)=Au\left(t\right)$, t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) $u\left(t\right)=p\left(t\right)+A{ʃ}_0^{t}a(t-s)u\left(s\right)ds$, t ∈ ℝ, where $a\left(·\right)\in L_{loc}^1\left(ℝ\right)$ and p(·) is a vector-valued polynomial of the form $p\left(t\right)={\sum }_{j=0}^{n}1/(j!)x_j{t}^j$ for some elements $x_j\in E$. We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension of a theorem...