On the spectral properties of differential operators with unbounded operator coefficients determined by a linear relation
Let \(X\) be a Banach space over \(\mathbb{R}\) or \(\mathbb{C}\) and let \({\mathcal{U}}: [a,b]\times[a,b]\to \text{End}(X)\) a map defining a family of (strongly continuous) evolution operators on \([a,b]\). If \(\Gamma\) is a linear relation on \(X,\) the equation \(x(t)={\mathcal{U}}(t,a)x(a)+\int_a^t{\mathcal{U}}(t,s)f(s)\,ds\) defines an operator \({\mathcal{L}}_{\Gamma}x=f\) for all continuous functions \(x:[a,b]\to X\) such that \((x(a),x(b))\in \Gamma\). In this interesting work, the author shows that, if the operator \({\mathcal{L}}_{\Gamma}\) has one of ten given spectral properties, then the operator \(\Gamma-{\mathcal{U}}(b,a)\) has the same spectral properties. In particular, he shows that, if one of them has the Fredholm property, then the other has the same property.
scientific article; zbMATH DE number 5980767
scientific article