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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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 $ E$ is a sequentially complete locally convex Hausdorff space and $ {(L(t))_{0 \leqslant t < \infty }}$ is a locally equicontinuous semigroup of class $ ({C_0})$ in $ E$ with generator $ S$ and if $ x \in E$ (not necessarily belonging to the domain of $ S$) then the function $ u(t) = L(t)x$ is a solution, in a generalized sense, of the initial value problem $ u'(t) = Su(t)$, $ u(0) = x$, and that such a generalized solution is unique. It should be noted here that $ u(t)$ may fail to belong to the domain of $ S$ so we must assign a suitable meaning to the expression $ Su(t)$.

References [Enhancements On Off] (What's this?)

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  • [3] Kôsaku Yosida, Functional analysis, 5th ed., Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Band 123. MR 0500055

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Article copyright: © Copyright 1983 American Mathematical Society