Cauchy Integrals Method in the Study of Perturbations of Operators

Sammanfattning: We show that functions which are analytic on the open unit disc and fulfil the Hölder condition of order r  there, where r lies in the interval (0,1), are operator Hölder of order r on the set of all linear contractions on a Hilbert space. Further, it is known that analytic Lipschitz functions on the unit disc need not be operator Lipschitz. We show that under a certain additional integral condition, these functions are operator Lipschitz. The two results are shown by tools from operator theory including the Spectral theorem and dilations of contractions. We also solve a problem related to theory of dilations which was arisen on a mathematical question- and answer site. More specificaly we show that, for a certain operator-valued polynomial, the von Neumann inequality is false.