  We generalize our results of \cite{AP2} and \cite{AP3} to the case of maximal
dissipative operators. We obtain sharp conditions on a function analytic in the
upper half-plane to be operator Lipschitz. We also show that a H\"older
function of order $\a$, $0<\a<1$, that is analytic in the upper half-plane must
be operator H\"older of order $\a$. Then we generalize these results to higher
order operator differences. We obtain sharp conditions for the existence of
operator derivatives and express operator derivatives in terms of multiple
operator integrals with respect to semi-spectral measures. Finally, we obtain
sharp estimates in the case of perturbations of Schatten-von Neumann class
$\bS_p$ and obtain analogs of all the results for commutators and
quasicommutators. Note that the proofs in the case of dissipative operators are
considerably more complicated than the proofs of the corresponding results for
self-adjoint operators, unitary operators, and contractions that were obtained
earlier in \cite{AP2}, \cite{AP3}, and \cite{Pe6}.

Functional Analysis;Complex Variables;Spectral Theory