Closable operators and semigroups
Author: Neil Falkner
Journal: Proc. Amer. Math. Soc. 87 (1983), 107-110
MSC: Primary 47D05; Secondary 34G10, 47A05
MathSciNet review: 677243
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Abstract: We show that a linear operator is closable iff it is possible to weaken the topology on its range in a certain nice way so as to render the operator continuous. We apply this result to show that if is a sequentially complete locally convex Hausdorff space and is a locally equicontinuous semigroup of class in with generator and if (not necessarily belonging to the domain of ) then the function is a solution, in a generalized sense, of the initial value problem , , and that such a generalized solution is unique. It should be noted here that may fail to belong to the domain of so we must assign a suitable meaning to the expression .
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