  We consider the abstract Cauchy problem x'=Ax, x(0)=x_0\in D(A) for linear
operators A on a Banach space X. We prove uniqueness of the (local) solution of
this problem for a natural class of operators A. Moreover, we establish that
the solution x(\cdot) can be represented as a limit of sequence F(t/n)^{n} as
n\to\infty in the weak operator topology, where a function F:[0,\infty)\to L(X)
satisfies F'(0)y=Ay, y\in D(A). As a consequence, we deduce necessary and
sufficient conditions that a linear operator C is closable and its closure is a
generator of C_0-semigroup. We also obtain some criteria for the sum of two
generators of C_0-semigroups to be a generator of C_0-semigroup such that the
Trotter formula is valid.
