Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators

Aleksei Aleksandrov; Fedor Nazarov; Vladimir Peller

doi:10.1016/j.crma.2015.05.005

Harmonic analysis/Functional analysis

Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators

Presented by: Gilles Pisier

Aleksei Aleksandrov ¹; Fedor Nazarov ²; Vladimir Peller ³

¹ Saint Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 Saint Petersburg, Russia
² Department of Mathematics, Kent State University, Kent, OH 44242, USA
³ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 723-728.

Abstract
Résumé

We study perturbations of functions $f (A, B)$ of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class $B_{\infty, 1}^{1} (R^{2})$ , then we have the following Lipschitz-type estimate in the Schatten–von Neumann norm $S_{p}$ , $1 \leq p \leq 2$ : ${‖ f (A_{1}, B_{1}) - f (A_{2}, B_{2}) ‖}_{S_{p}} \leq const ({‖ A_{1} - A_{2} ‖}_{S_{p}} + {‖ B_{1} - B_{2} ‖}_{S_{p}})$ . However, the condition $f \in B_{\infty, 1}^{1} (R^{2})$ does not imply the Lipschitz-type estimate in $S_{p}$ with $p > 2$ . The main tool is Schatten–von Neumann norm estimates for triple operator integrals.

Nous examinons les perturbations de fonctions $f (A, B)$ d'opérateurs auto-adjoints A et B qui ne commutent pas. De telles fonctions peuvent être définies en termes d'intégrales doubles opératorielles. Pour f dans l'espace de Besov $B_{\infty, 1}^{1} (R^{2})$ , nous obtenons l'estimation lipschitzienne en norme de Schatten–von Neumann $S_{p}$ , $1 \leq p \leq 2$ : ${‖ f (A_{1}, B_{1}) - f (A_{2}, B_{2}) ‖}_{S_{p}} \leq const ({‖ A_{1} - A_{2} ‖}_{S_{p}} + {‖ B_{1} - B_{2} ‖}_{S_{p}})$ . Par ailleurs, la condition $f \in B_{\infty, 1}^{1} (R^{2})$ n'implique pas l'estimation lipschitzienne en norme de $S_{p}$ pour $p > 2$ . L'outil principal consiste en l'estimation d'intégrales triples opératorielles dans les normes de $S_{p}$ .

Received: 2015-04-06
Accepted: 2015-05-22
Published online: 2015-06-17

DOI: 10.1016/j.crma.2015.05.005

Author's affiliations:

Aleksei Aleksandrov ¹; Fedor Nazarov ²; Vladimir Peller ³

¹ Saint Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 Saint Petersburg, Russia
² Department of Mathematics, Kent State University, Kent, OH 44242, USA
³ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

@article{CRMATH_2015__353_8_723_0,
     author = {Aleksei Aleksandrov and Fedor Nazarov and Vladimir Peller},
     title = {Triple operator integrals in {Schatten{\textendash}von} {Neumann} norms and functions of perturbed noncommuting operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {723--728},
     publisher = {Elsevier},
     volume = {353},
     number = {8},
     year = {2015},
     doi = {10.1016/j.crma.2015.05.005},
     language = {en},
}

TY  - JOUR
AU  - Aleksei Aleksandrov
AU  - Fedor Nazarov
AU  - Vladimir Peller
TI  - Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 723
EP  - 728
VL  - 353
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2015.05.005
LA  - en
ID  - CRMATH_2015__353_8_723_0
ER  -

%0 Journal Article
%A Aleksei Aleksandrov
%A Fedor Nazarov
%A Vladimir Peller
%T Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators
%J Comptes Rendus. Mathématique
%D 2015
%P 723-728
%V 353
%N 8
%I Elsevier
%R 10.1016/j.crma.2015.05.005
%G en
%F CRMATH_2015__353_8_723_0

Aleksei Aleksandrov; Fedor Nazarov; Vladimir Peller. Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators. Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 723-728. doi : 10.1016/j.crma.2015.05.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.05.005/

References
Cited by

[1] A.B. Aleksandrov; V.V. Peller Operator Hölder–Zygmund functions, Adv. Math., Volume 224 (2010), pp. 910-966

[2] A.B. Aleksandrov; V.V. Peller; D. Potapov; F. Sukochev Functions of normal operators under perturbations, Adv. Math., Volume 226 (2011), pp. 5216-5251

[3] A.B. Aleksandrov; F.L. Nazarov; V.V. Peller Functions of perturbed noncommuting self-adjoint operators, C. R. Acad. Sci. Paris, Ser. I, Volume 353 (2015), pp. 209-214

[4] J. Bergh; J. Löfström Interpolation Spaces, Springer-Verlag, Berlin, 1976

[5] M.S. Birman; M.Z. Solomyak Double Stieltjes operator integrals, Probl. Math. Phys., Leningrad. Univ., Volume 1 (1966), pp. 33-67 (in Russian). English transl.: Top. Math. Phys. 1 (1967) 25–54, Consultants Bureau Plenum Publishing Corporation, New York

[6] Yu.B. Farforovskaya The connection of the Kantorovich–Rubinshtein metric for spectral resolutions of selfadjoint operators with functions of operators, Vestn. Leningr. Univ., Math., Volume 19 (1968), pp. 94-97 (in Russian)

[7] K. Juschenko; I.G. Todorov; L. Turowska Multidimensional operator multipliers, Trans. Amer. Math. Soc., Volume 361 (2009), pp. 4683-4720

[8] F.L. Nazarov; V.V. Peller Functions of n-tuples of commuting self-adjoint operators, J. Funct. Anal., Volume 266 (2014), pp. 5398-5428

[9] J. Peetre New Thoughts on Besov Spaces, Duke University Press, Durham, NC, USA, 1976

[10] V.V. Peller Hankel operators in the theory of perturbations of unitary and self-adjoint operators, Funkc. Anal. Prilozh., Volume 19 (1985) no. 2, pp. 37-51 (in Russian). English transl.: Funct. Anal. Appl. 19 (1985) 111–123

[11] V.V. Peller Hankel operators in the perturbation theory of unbounded self-adjoint operators, Analysis and Partial Differential Equations, Lecture Notes Pure Appl. Math., vol. 122, Dekker, New York, 1990, pp. 529-544

[12] V.V. Peller Hankel Operators and Their Applications, Springer-Verlag, New York, 2003

[13] V.V. Peller Multiple operator integrals and higher operator derivatives, J. Funct. Anal., Volume 233 (2006), pp. 515-544

[14] G. Pisier Introduction to Operator Space Theory, London Math. Soc. Lecture Notes Ser., vol. 294, Cambridge University Press, Cambridge, UK, 2003

Cited by Sources:

Comments - Policy