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

Aleksandrov, Aleksei; Nazarov, Fedor; Peller, Vladimir

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
[Intégrales triples opératorielles en normes de Schatten–von Neumann et fonctions d'opérateurs perturbés ne commutant pas]

Présenté par : Gilles Pisier

Aleksandrov, Aleksei ¹ ; Nazarov, Fedor ² ; Peller, Vladimir 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, Tome 353 (2015) no. 8, pp. 723-728.

Résumé
Abstract

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 $S_{p}$ .

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.

Reçu le : 2015-04-06
Accepté le : 2015-05-22
Publié le : 2015-06-17

DOI : 10.1016/j.crma.2015.05.005

Affiliations des auteurs :

Aleksandrov, Aleksei ¹ ; Nazarov, Fedor ² ; Peller, Vladimir ³

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Aleksandrov, Aleksei; Nazarov, Fedor; Peller, Vladimir. Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators. Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 723-728. doi : 10.1016/j.crma.2015.05.005. http://archive.numdam.org/articles/10.1016/j.crma.2015.05.005/

Bibliographie
Cité par

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

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

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

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

[5] Birman, M.S.; Solomyak, M.Z. 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] Farforovskaya, Yu.B. 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] Juschenko, K.; Todorov, I.G.; Turowska, L. Multidimensional operator multipliers, Trans. Amer. Math. Soc., Volume 361 (2009), pp. 4683-4720

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

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

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

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

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

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