  The transfer operator associated to a flow (continuous time dynamical system)
is a one-parameter operator semigroup. We consider the operator-valued Laplace
transform of this one-parameter semigroup. Estimates on the Laplace transform
have be used in various settings in order to show the rate at which the flow
mixes. Here we consider the case of exponential mixing or rapid mixing (super
polynomial). We develop the operator theory framework amenable to this setting
and show that the same estimates may be used to produce results, in terms of
the operators, which go beyond the results for the rate of mixing. Such results
are useful for obtaining other statistical properties of the dynamical system.
