Mathematical analysis/Functional analysis
Functions of perturbed noncommuting self-adjoint operators
Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 209-214.

We consider 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 $B∞,11(R2)$, then we have the following Lipschitz-type estimate in the trace norm: $‖f(A1,B1)−f(A2,B2)‖S1≤const(‖A1−A2‖S1+‖B1−B2‖S1)$. However, the condition $f∈B∞,11(R2)$ does not imply the Lipschitz-type estimate in the operator norm.

Nous considérons les 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 $B∞,11(R2)$, nous obtenons l'estimation lipschitzienne en norme trace : $‖f(A1,B1)−f(A2,B2)‖S1≤const(‖A1−A2‖S1+‖B1−B2‖S1)$. Par ailleurs, la condition $f∈B∞,11(R2)$ n'implique pas l'estimation lip-schitzienne en norme opératorielle.

Accepted:
Published online:
DOI: 10.1016/j.crma.2014.12.005

Aleksei Aleksandrov 1; Fedor Nazarov 2; Vladimir Peller 3

1 St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St-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_3_209_0,
author = {Aleksei Aleksandrov and Fedor Nazarov and Vladimir Peller},
title = {Functions of perturbed noncommuting self-adjoint operators},
journal = {Comptes Rendus. Math\'ematique},
pages = {209--214},
publisher = {Elsevier},
volume = {353},
number = {3},
year = {2015},
doi = {10.1016/j.crma.2014.12.005},
language = {en},
}
TY  - JOUR
AU  - Aleksei Aleksandrov
AU  - Fedor Nazarov
TI  - Functions of perturbed noncommuting self-adjoint operators
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 209
EP  - 214
VL  - 353
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2014.12.005
LA  - en
ID  - CRMATH_2015__353_3_209_0
ER  -
%0 Journal Article
%A Aleksei Aleksandrov
%A Fedor Nazarov
%T Functions of perturbed noncommuting self-adjoint operators
%J Comptes Rendus. Mathématique
%D 2015
%P 209-214
%V 353
%N 3
%I Elsevier
%R 10.1016/j.crma.2014.12.005
%G en
%F CRMATH_2015__353_3_209_0
Aleksei Aleksandrov; Fedor Nazarov; Vladimir Peller. Functions of perturbed noncommuting self-adjoint operators. Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 209-214. doi : 10.1016/j.crma.2014.12.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.12.005/

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

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

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

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

[7] 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, pp. 111-123

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

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

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

[11] G. Pisier Introduction to Operator Space Theory, Lond. Math. Soc. Lect. Note Ser., vol. 294, Cambridge University Press, Cambridge, UK, 2003

Cited by Sources: