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.
UDC:
517.983.23
MSC: Primary
47D06,
34G10; Secondary
47D03,
47D60 Received: 31.03.2008 and 16.12.2008
DOI:
10.4213/sm5097
Fulltext:
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