TY - JOUR

T1 - Functions of perturbed dissipative operators

AU - Peller, V. V.

AU - Александров, Алексей Борисович

PY - 2012

Y1 - 2012

N2 - 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 Ḧolder function of order α, 0 < α < 1, that is analytic in the upper half-plane must be operator Ḧolder of order α. 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 Sp 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.

AB - 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 Ḧolder function of order α, 0 < α < 1, that is analytic in the upper half-plane must be operator Ḧolder of order α. 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 Sp 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.

KW - Besov spaces

KW - Continuity moduli

KW - Dissipative operators

KW - Hölder-Zygmund spaces

KW - Perturbations of operators

KW - Schatten-von Neumann classes

UR - http://www.scopus.com/inward/record.url?scp=84868028676&partnerID=8YFLogxK

U2 - 10.1090/S1061-0022-2012-01194-5

DO - 10.1090/S1061-0022-2012-01194-5

M3 - Article

AN - SCOPUS:84868028676

VL - 23

SP - 209

EP - 238

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 2

ER -