Comptes Rendus
Mathematical Analysis/Functional Analysis
Functions of perturbed tuples of self-adjoint operators
[Fonctions dʼuplets dʼopérateurs autoadjoints perturbés]
Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 349-354.

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,11(Rn), alors elle est lipschitzienne opératorielle. En outre, nous montrons que si f appartient à lʼespace de Hölder dʼordre α, alors f(A1,,An)f(B1,,Bn)constmax1jnAjBjα por tous n-uplets (A1,,An) et (B1,,Bn) dʼopérateurs auto-adjoints qui commutent. Nous considérons aussi le cas de module de continuité arbitraire et le cas où les opérateurs AjBj appartiennent à lʼespace de Schatten–von Neumann Sp.

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,11(Rn), then f is operator Lipschitz and we show that if f satisfies a Hölder condition of order α, then f(A1,,An)f(B1,,Bn)constmax1jnAjBjα for all n-tuples of commuting self-adjoint operators (A1,,An) and (B1,,Bn). We also consider the case of arbitrary moduli of continuity and the case when the operators AjBj belong to the Schatten–von Neumann class Sp.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.04.010

Fedor Nazarov 1 ; Vladimir Peller 2

1 Department of Mathematics, Kent State University, Kent, OH 44242, USA
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
@article{CRMATH_2012__350_7-8_349_0,
     author = {Fedor Nazarov and Vladimir Peller},
     title = {Functions of perturbed tuples of self-adjoint operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {349--354},
     publisher = {Elsevier},
     volume = {350},
     number = {7-8},
     year = {2012},
     doi = {10.1016/j.crma.2012.04.010},
     language = {en},
}
TY  - JOUR
AU  - Fedor Nazarov
AU  - Vladimir Peller
TI  - Functions of perturbed tuples of self-adjoint operators
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 349
EP  - 354
VL  - 350
IS  - 7-8
PB  - Elsevier
DO  - 10.1016/j.crma.2012.04.010
LA  - en
ID  - CRMATH_2012__350_7-8_349_0
ER  - 
%0 Journal Article
%A Fedor Nazarov
%A Vladimir Peller
%T Functions of perturbed tuples of self-adjoint operators
%J Comptes Rendus. Mathématique
%D 2012
%P 349-354
%V 350
%N 7-8
%I Elsevier
%R 10.1016/j.crma.2012.04.010
%G en
%F CRMATH_2012__350_7-8_349_0
Fedor Nazarov; Vladimir Peller. Functions of perturbed tuples of self-adjoint operators. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 349-354. doi : 10.1016/j.crma.2012.04.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.04.010/

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

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

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

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

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

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

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

[10] 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)

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

[13] V.V. Peller 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] 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, Dekker, New York, 1990, pp. 529-544

Cité par Sources :

Commentaires - Politique