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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Functions of perturbed dissipative operators


Authors: A. B. Aleksandrov and V. V. Peller
Translated by: the authors
Original publication: Algebra i Analiz, tom 23 (2011), nomer 2.
Journal: St. Petersburg Math. J. 23 (2012), 209-238
MSC (2010): Primary 47A56; Secondary 47B44
Published electronically: January 23, 2012
MathSciNet review: 2841671
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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.


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Additional Information

A. B. Aleksandrov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
Email: alex@pdmi.ras.ru

V. V. Peller
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: peller@math.msu.edu

DOI: http://dx.doi.org/10.1090/S1061-0022-2012-01194-5
PII: S 1061-0022(2012)01194-5
Keywords: Dissipative operators, perturbations of operators, Schatten–von Neumann classes, Hölder–Zygmund spaces, Besov spaces, continuity moduli
Received by editor(s): September 22, 2010
Published electronically: January 23, 2012
Additional Notes: The first author was partially supported by RFBR grant 08-01-00358-a.
The second author was partially supported by NSF grant DMS 1001844 and by ARC grant.
Dedicated: To the memory of M. Sh. Birman
Article copyright: © Copyright 2012 American Mathematical Society