Comptes Rendus
Mathematical Analysis/Functional Analysis
Lipschitz functions of perturbed operators
[Fonctions lipschitziennes d'opérateurs perturbés]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 857-862.

Nous démontrons que si f est une fonction lipschitzienne, A et B des opérateurs autoadjoints tels que rank(AB)=1, alors f(A)f(B)S1,, c'est-à-dire sj(AB)const(1+j)1. Si AB est dans la classe S1 des opérateurs à trace, nous montrons que f(A)f(B)SΩ, c'est-à-dire j=0nsj(f(A)f(B))constlog(2+n). Plus généralement, pour une fonction lipschitzienne f et pour des mesures spectrales E1 et E2, considérons l'intégrale double opératorielle Q=(f(x)f(y))(xy)1dE1(x)TdE2(y). Nous montrons que si TS1, alors QSΩ et si rankT=1, alors QS1,. Finalement, si T appartient à l'idéal de Matsaev Sω, 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(AB)=1, then f(A)f(B) belongs to the weak space S1,, i.e., sj(AB)const(1+j)1. We deduce from this result that if AB belongs to the trace class S1 and f is Lipschitz, then f(A)f(B)SΩ, i.e., j=0nsj(f(A)f(B))constlog(2+n). We also obtain more general results about the behavior of double operator integrals of the form Q=(f(x)f(y))(xy)1dE1(x)TdE2(y), where E1 and E2 are spectral measures. We show that if TS1, then QSΩ and if rankT=1, then QS1,. Finally, if T belongs to the Matsaev ideal Sω, then Q is a compact operator.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.05.003

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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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/

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

[5] Yu.B. Farforovskaya 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] V.V. Peller 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] 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

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

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