Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems

Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem ${\displaystyle u^{\left(n\right)}\left(t\right)=Au\left(t\right)}$, t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) ${\displaystyle u\left(t\right)=p\left(t\right)+A{ʃ}_0^{t}a(t-s)u\left(s\right)ds}$, t ∈ ℝ, where ${\displaystyle a(·)\in L_{loc}^1(ℝ)}$ and p(·) is a vector-valued polynomial of the form ${\displaystyle p\left(t\right)={\sum }_{j=0}^n\frac{1}{j!}{x}_j{t}^j}$ for some elements ${\displaystyle x_j\in E}$. 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 of Pólya in complex analysis is obtained and applied to the individual "Ax = 0" and "Tx = x" problem.