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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.

##### MSC:
 34G10 Linear differential equations in abstract spaces 47D06 One-parameter semigroups and linear evolution equations 34K30 Functional-differential equations in abstract spaces
##### Keywords:
evolution semigroups; commuting operators; solvability
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##### References:
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