Analogues of Chernoff's theorem and the Lie-Trotter theorem

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.
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