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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Closable operators and semigroups
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by Neil Falkner PDF
Proc. Amer. Math. Soc. 87 (1983), 107-110 Request permission

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
  • Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
  • K. Yosida, Lectures on semi-group theory and its application to Cauchy’s problem in partial differential equations, Tata Institute of Fundamental Research, Bombay, 1957.
  • Kôsaku Yosida, Functional analysis, 5th ed., Grundlehren der Mathematischen Wissenschaften, Band 123, Springer-Verlag, Berlin-New York, 1978. MR 0500055
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 87 (1983), 107-110
  • MSC: Primary 47D05; Secondary 34G10, 47A05
  • DOI: https://doi.org/10.1090/S0002-9939-1983-0677243-5
  • MathSciNet review: 677243