Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems
Volume 116 / 1995
Studia Mathematica 116 (1995), 23-41
DOI: 10.4064/sm-116-1-23-41
Abstract
Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem $u^{(n)}(t) = Au(t)$, t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) $u(t) = p(t) + A ʃ_{0}^{t} a(t-s)u(s)ds$, t ∈ ℝ, where $a(·) ∈ L_{loc}^{1}(ℝ)$ and p(·) is a vector-valued polynomial of the form $p(t) = ∑_{j=0}^n 1/(j!) x_j t^j$ for some elements $x_j ∈ E$. 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.
