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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...

Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function ${\phi }_a\left(t\right):=\phi \left({\alpha }_{t}a\right)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum ${\sigma }_w*\left({\phi }_a\right)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define ${Ʌ}_{\phi }^a$ to be the union of all...