Functions of perturbed tuples of self-adjoint operators

Nazarov, Fedor; Peller, Vladimir

doi:10.1016/j.crma.2012.04.010

Mathematical Analysis/Functional Analysis

Functions of perturbed tuples of self-adjoint operators
[Fonctions dʼuplets dʼopérateurs autoadjoints perturbés]

Présenté par : Gilles Pisier

Nazarov, Fedor ¹ ; Peller, Vladimir ²

¹ Department of Mathematics, Kent State University, Kent, OH 44242, USA
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Tome 350 (2012) no. 7-8, pp. 349-354.

Résumé
Abstract

Dans cette Note nous généralisons des résultats de Aleksandrov et Peller (2010) [2,3], Aleksandrov et al. (2011) [6], Peller (1985) [13], Peller (1990) [14] en cas de fonctions dʼopérateurs auto-adjoints et dʼopérateurs normaux. Nous considérons le problème similaire pour les fonctions de n-uplets dʼopérateurs auto-adjoints qui commutent. En particulier, nous démontrons que si f est une fonction de la classe de Besov $B_{\infty, 1}^{1} (R^{n})$ , alors elle est lipschitzienne opératorielle. En outre, nous montrons que si f appartient à lʼespace de Hölder dʼordre α, alors $‖ f (A_{1}, \dots, A_{n}) - f (B_{1}, \dots, B_{n}) ‖ ⩽ const \max_{1 ⩽ j ⩽ n} ‖ A_{j} - B_{j} ‖^{α}$ por tous n-uplets $(A_{1}, \dots, A_{n})$ et $(B_{1}, \dots, B_{n})$ dʼopérateurs auto-adjoints qui commutent. Nous considérons aussi le cas de module de continuité arbitraire et le cas où les opérateurs $A_{j} - B_{j}$ appartiennent à lʼespace de Schatten–von Neumann $S_{p}$ .

We generalize earlier results of Aleksandrov and Peller (2010) [2,3], Aleksandrov et al. (2011) [6], Peller (1985) [13], Peller (1990) [14] to the case of functions of n-tuples of commuting self-adjoint operators. In particular, we prove that if a function f belongs to the Besov space $B_{\infty, 1}^{1} (R^{n})$ , then f is operator Lipschitz and we show that if f satisfies a Hölder condition of order α, then $‖ f (A_{1}, \dots, A_{n}) - f (B_{1}, \dots, B_{n}) ‖ ⩽ const \max_{1 ⩽ j ⩽ n} ‖ A_{j} - B_{j} ‖^{α}$ for all n-tuples of commuting self-adjoint operators $(A_{1}, \dots, A_{n})$ and $(B_{1}, \dots, B_{n})$ . We also consider the case of arbitrary moduli of continuity and the case when the operators $A_{j} - B_{j}$ belong to the Schatten–von Neumann class $S_{p}$ .

Reçu le : 2012-03-07
Accepté le : 2012-04-10
Publié le : 2012-04-01

DOI : 10.1016/j.crma.2012.04.010

Affiliations des auteurs :

Nazarov, Fedor ¹ ; Peller, Vladimir ²

¹ Department of Mathematics, Kent State University, Kent, OH 44242, USA
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

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Nazarov, Fedor; Peller, Vladimir. Functions of perturbed tuples of self-adjoint operators. Comptes Rendus. Mathématique, Tome 350 (2012) no. 7-8, pp. 349-354. doi : 10.1016/j.crma.2012.04.010. http://www.numdam.org/articles/10.1016/j.crma.2012.04.010/

Bibliographie
Cité par

[1] Aleksandrov, A.B.; Peller, V.V. Functions of perturbed operators, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009), pp. 483-488

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

[3] Aleksandrov, A.B.; Peller, V.V. Functions of operators under perturbations of class $S_{p}$ , J. Funct. Anal., Volume 258 (2010), pp. 3675-3724

[4] Aleksandrov, A.B.; Peller, V.V. Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities, Indiana Univ. Math. J., Volume 59 (2010) no. 4, pp. 1451-1490

[5] Aleksandrov, A.B.; Peller, V.V.; Potapov, D.; Sukochev, F. Functions of perturbed normal operators, C. R. Acad. Sci. Paris, Ser I, Volume 348 (2010), pp. 553-558

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

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

[8] Birman, M.S.; Solomyak, M.Z. Double Stieltjes operator integrals. II, Problems of Math. Phys., vol. 2, Leningrad. Univ., New York, 1967, pp. 26-60 (in Russian); English transl.:, Top. Math. Phys., vol. 2, 1968, Consultants Bureau Plenum Publishing Corporation, pp. 19-46

[9] Birman, M.S.; Solomyak, M.Z. Double Stieltjes operator integrals. III, Problems of Math. Phys., vol. 6, Leningrad. Univ., 1973, pp. 27-53 (in Russian)

[10] Farforovskaya, Yu.B. 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)

[11] E. Kissin, V.S. Shulman, Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations, in press.

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

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

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

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