Lipschitz functions of perturbed operators

Fedor Nazarov; Vladimir Peller

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

Fedor Nazarov ¹ ; Vladimir Peller 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, Volume 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 :

Fedor Nazarov ¹ ; Vladimir Peller ²

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

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     title = {Lipschitz functions of perturbed operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {857--862},
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     number = {15-16},
     year = {2009},
     doi = {10.1016/j.crma.2009.05.003},
     language = {en},
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Fedor Nazarov; Vladimir Peller. Lipschitz functions of perturbed operators. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 857-862. doi : 10.1016/j.crma.2009.05.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.05.003/

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