  In this paper we prove the existence of global classical solutions to
continuous coagulation-fragmentation equations with unbounded coefficients
under the sole assumption that the coagulation rate is dominated by a power of
the fragmentation rate, thus improving upon a number of recent results by not
requiring any polynomial growth bound for either rate. This is achieved by
proving a new result on the analyticity of the fragmentation semigroup and then
using its regularizing properties to prove the local and then, under a stronger
assumption, the global classical solvability of the coagulation-fragmentation
equation considered as a semilinear perturbation of the linear fragmentation
equation. Furthermore, we show that weak solutions of the
coagulation--fragmentation equation, obtained by the weak compactness method,
coincide with the classical local in time solutions provided the latter exist.
