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

This paper is concerned with the abstract Cauchy problem x = Ax, x(0) = x(0) is an element of D(A), where A is a densely defined linear operator on a Banach space X. It is proved that a solution x(.) of this problem can be represented as the weak limit lim(n ->infinity) {F(t/n)(n)xo}, where the function IF: [0, infinity) -> L(X) satisfies the equality F'(0)y = Ay, y E D(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 C to be closable and for its closure to be the generator of a Co-semigroup. Also, we obtain new criteria for the sum of two generators of Co-semigroup to be the generator of a Co-semigroup and for the Lie-trotter formula to hold.