Lipschitz functions of perturbed operators

Nazarov, Fedor; Peller, Vladimir

doi:10.1016/j.crma.2009.05.003

Mathematical Analysis/Functional Analysis

Lipschitz functions of perturbed operators
[Fonctions lipschitziennes d'opérateurs perturbés]

Présenté par : Gilles Pisier

Nazarov, Fedor ¹ ; Peller, Vladimir ²

¹ Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Tome 347 (2009) no. 15-16, pp. 857-862.

Résumé
Abstract

Nous démontrons que si f est une fonction lipschitzienne, A et B des opérateurs autoadjoints tels que $rank (A - B) = 1$ , alors $f (A) - f (B) \in S_{1, \infty}$ , c'est-à-dire $s_{j} (A - B) ⩽ const {(1 + j)}^{- 1}$ . Si $A - B$ est dans la classe $S_{1}$ des opérateurs à trace, nous montrons que $f (A) - f (B) \in S_{Ω}$ , c'est-à-dire $\sum_{j = 0}^{n} s_{j} (f (A) - f (B)) ⩽ const \log (2 + n)$ . Plus généralement, pour une fonction lipschitzienne f et pour des mesures spectrales $E_{1}$ et $E_{2}$ , considérons l'intégrale double opératorielle $Q = \int \int (f (x) - f (y)) {(x - y)}^{- 1} d E_{1} (x) T d E_{2} (y)$ . Nous montrons que si $T \in S_{1}$ , alors $Q \in S_{Ω}$ et si $rank T = 1$ , alors $Q \in S_{1, \infty}$ . Finalement, si T appartient à l'idéal de Matsaev $S_{ω}$ , alors Q est un opérateur compact.

We prove that if f is a Lipschitz function on $R$ , and A and B are self-adjoint operators such that $rank (A - B) = 1$ , then $f (A) - f (B)$ belongs to the weak space $S_{1, \infty}$ , i.e., $s_{j} (A - B) ⩽ const {(1 + j)}^{- 1}$ . We deduce from this result that if $A - B$ belongs to the trace class $S_{1}$ and f is Lipschitz, then $f (A) - f (B) \in S_{Ω}$ , i.e., $\sum_{j = 0}^{n} s_{j} (f (A) - f (B)) ⩽ const \log (2 + n)$ . We also obtain more general results about the behavior of double operator integrals of the form $Q = \int \int (f (x) - f (y)) {(x - y)}^{- 1} d E_{1} (x) T d E_{2} (y)$ , where $E_{1}$ and $E_{2}$ are spectral measures. We show that if $T \in S_{1}$ , then $Q \in S_{Ω}$ and if $rank T = 1$ , then $Q \in S_{1, \infty}$ . Finally, if T belongs to the Matsaev ideal $S_{ω}$ , then Q is a compact operator.

Reçu le : 2009-05-04
Accepté le : 2009-05-13
Publié le : 2009-06-23

DOI : 10.1016/j.crma.2009.05.003

Affiliations des auteurs :

Nazarov, Fedor ¹ ; Peller, Vladimir ²

¹ Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

@article{CRMATH_2009__347_15-16_857_0,
     author = {Nazarov, Fedor and Peller, Vladimir},
     title = {Lipschitz functions of perturbed operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {857--862},
     publisher = {Elsevier},
     volume = {347},
     number = {15-16},
     year = {2009},
     doi = {10.1016/j.crma.2009.05.003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.05.003/}
}

TY  - JOUR
AU  - Nazarov, Fedor
AU  - Peller, Vladimir
TI  - Lipschitz functions of perturbed operators
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 857
EP  - 862
VL  - 347
IS  - 15-16
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2009.05.003/
DO  - 10.1016/j.crma.2009.05.003
LA  - en
ID  - CRMATH_2009__347_15-16_857_0
ER  -

%0 Journal Article
%A Nazarov, Fedor
%A Peller, Vladimir
%T Lipschitz functions of perturbed operators
%J Comptes Rendus. Mathématique
%D 2009
%P 857-862
%V 347
%N 15-16
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2009.05.003/
%R 10.1016/j.crma.2009.05.003
%G en
%F CRMATH_2009__347_15-16_857_0

Nazarov, Fedor; Peller, Vladimir. Lipschitz functions of perturbed operators. Comptes Rendus. Mathématique, Tome 347 (2009) no. 15-16, pp. 857-862. doi : 10.1016/j.crma.2009.05.003. http://www.numdam.org/articles/10.1016/j.crma.2009.05.003/

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

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

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

[5] Farforovskaya, Yu.B. An example of a Lipschitzian function of selfadjoint operators that yields a nonnuclear increase under a nuclear perturbation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 30 (1972), pp. 146-153 (in Russian)

[6] Peller, V.V. Metric properties of the averaging projection onto the set of Hankel matrices, Dokl. Akad. Nauk SSSR, Volume 278 (1984), pp. 271-285 (English transl. in: Soviet Math. Dokl., 30, 1984, pp. 362-368)

[7] 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

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

Cité par Sources :