Functions of perturbed dissipative operators

Abstract:

We generalize our earlier results 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 Hölder function of order $\alpha$, $0<\alpha <1$, that is analytic in the upper half-plane must be operator Hölder of order $\alpha$. More general results for arbitrary moduli of continuity will also be obtained. 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 $\boldsymbol {S}_p$ 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.