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Evolution semigroups and sums of commuting operators: A new approach to the admissibility theory of function spaces. (English) Zbl 0966.34049
The authors study the equation \((*)\) \({d}/{dt}\) \(u(t) = Au(t) + f(t)\), where \(A\) is a generator of a \(C_{0}\)-semigroup on a Banach space \(X\), and are interested in which properties of the function \(f\) are inherited by the solution \(u\). To that purpose they consider the generator \(G\) of the evolution semigroup \(T(t)g(s) = e^{tA}g(s-t)\) on \(X\)-valued function spaces on \(\mathbb{R}\). Formally this generator is the sum of \({-d}/{dt}\) and the multiplication operator given by \(A\). They use spectral theory to find criteria for the solvability of \((*)\). The method and the results are also applied to higher-order and functional-differential equations.

34G10 Linear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
34K30 Functional-differential equations in abstract spaces
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