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
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