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Analogues of Chernoff's theorem and the Lie-Trotter theorem
A. Yu. Neklyudov M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
This paper is concerned with the abstract Cauchy problem $\dot x=\mathrm{A}x$, $x(0)=x_0\in\mathscr{D}(\mathrm{A})$, where $\mathrm{A}$ is a densely defined linear operator on a Banach space $\mathbf X$. It is proved that a solution $x(\,\cdot\,)$ of this problem can be represented as the weak limit
$\lim_{n\to\infty}\{\mathrm F(t/n)^nx_0\}$, where the function
$\mathrm F\colon[0,\infty)\mapsto\mathscr L(\mathrm X)$ satisfies the equality
$\mathrm F'(0)y=\mathrm{A}y$, $y\in\mathscr{D}(\mathrm{A})$,
for a natural class of operators. As distinct from Chernoff's theorem, the existence of a global solution to the Cauchy problem is not assumed. Based on this result, necessary and sufficient conditions are found
for the linear operator $\mathrm{C}$ to be closable and for its closure to be the generator of a $C_0$-semigroup. Also, we obtain new criteria for the sum of two generators of $C_0$-semigroups to be the generator of a $C_0$-semigroup and for the Lie-Trotter formula to hold.
Bibliography: 13 titles.
Keywords:
Chernoff's theorem; Lie-Trotter theorem; semigroup.
Received: 31.03.2008 and 16.12.2008
Citation:
A. Yu. Neklyudov, “Analogues of Chernoff's theorem and the Lie-Trotter theorem”, Mat. Sb., 200:10 (2009), 81–106; Sb. Math., 200:10 (2009), 1495–1519
Linking options:
https://www.mathnet.ru/eng/sm5097https://doi.org/10.1070/SM2009v200n10ABEH004047 https://www.mathnet.ru/eng/sm/v200/i10/p81
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Abstract page: | 602 | Russian version PDF: | 229 | English version PDF: | 5 | References: | 37 | First page: | 14 |
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