Weak type operator Lipschitz and commutator estimates for commuting tuples

Caspers, Martijn; Sukochev, Fedor; Zanin, Dmitriy

doi:10.5802/aif.3195

Weak type operator Lipschitz and commutator estimates for commuting tuples [ Estimées de type faible pour les fonctions operateur-Lipschitz et les commutateurs de plusieurs variables ]

Caspers, Martijn ; Sukochev, Fedor ; Zanin, Dmitriy

Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1643-1669.

Résumé
Abstract

Soit $f : ℝ^{d} \to ℝ$ une fonction Lipschitzienne. Si $B$ est un opérateur borné auto-adjoint et si ${A_{k}}_{k = 1}^{d}$ sont des opérateurs bornés auto-adjoints qui commutent et tels que $[A_{k}, B] \in L_{1} (H),$ alors

∥ [f (A_{1}, \dots, A_{d}), B] ∥_{1, \infty} \leq c (d) ∥ \nabla (f) ∥_{\infty} max_{1 \leq k \leq d} {∥ [A_{k}, B] ∥}_{1},

où $c (d)$ est une constante indépendante de $f$ , $ℳ$ et $A, B$ et ${∥ \cdot ∥}_{1, \infty}$ désigne la norme $L_{1}$ -faible.

Si ${X_{k}}_{k = 1}^{d}$ (respectivement ${Y_{k}}_{k = 1}^{d}$ ) sont des opérateurs bornés qui commutent et tels que $X_{k} - Y_{k} \in L_{1} (H),$ alors

∥ f (X_{1}, \dots, X_{d}) - f (Y_{1}, \dots, Y_{d}) ∥_{1, \infty} \leq c (d) ∥ \nabla (f) ∥_{\infty} max_{1 \leq k \leq d} {∥ X_{k} - Y_{k} ∥}_{1} .

Let $f : ℝ^{d} \to ℝ$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if ${A_{k}}_{k = 1}^{d}$ are commuting bounded self-adjoint operators such that $[A_{k}, B] \in L_{1} (H),$ then

∥ [f (A_{1}, \dots, A_{d}), B] ∥_{1, \infty} \leq c (d) ∥ \nabla (f) ∥_{\infty} max_{1 \leq k \leq d} {∥ [A_{k}, B] ∥}_{1},

where $c (d)$ is a constant independent of $f$ , $ℳ$ and $A, B$ and ${∥ \cdot ∥}_{1, \infty}$ denotes the weak $L_{1}$ -norm.

If ${X_{k}}_{k = 1}^{d}$ (respectively, ${Y_{k}}_{k = 1}^{d}$ ) are commuting bounded self-adjoint operators such that $X_{k} - Y_{k} \in L_{1} (H),$ then

∥ f (X_{1}, \dots, X_{d}) - f (Y_{1}, \dots, Y_{d}) ∥_{1, \infty} \leq c (d) ∥ \nabla (f) ∥_{\infty} max_{1 \leq k \leq d} {∥ X_{k} - Y_{k} ∥}_{1} .

Reçu le : 2017-03-09
Révisé le : 2017-09-21
Accepté le : 2017-11-07
Publié le : 2018-11-23

DOI : https://doi.org/10.5802/aif.3195
Classification : 47B10, 47L20, 47A30
Mots clés: Espaces

L_{p}

non commutatifs, estimées de commutateurs, théorie de Calderón–Zygmund

@article{AIF_2018__68_4_1643_0,
     author = {Caspers, Martijn and Sukochev, Fedor and Zanin, Dmitriy},
     title = {Weak type operator Lipschitz and commutator estimates for commuting tuples},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {4},
     year = {2018},
     pages = {1643-1669},
     doi = {10.5802/aif.3195},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2018__68_4_1643_0/}
}

Caspers, Martijn; Sukochev, Fedor; Zanin, Dmitriy. Weak type operator Lipschitz and commutator estimates for commuting tuples. Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1643-1669. doi : 10.5802/aif.3195. https://aif.centre-mersenne.org/item/AIF_2018__68_4_1643_0/

Bibliographie

[1] Aleksandrov, Aleksei; Peller, Vladimir; Potapov, Denis; Sukochev, Fedor Functions of normal operators under perturbations, Adv. Math., Tome 226 (2011) no. 6, pp. 5216-5251 | Zbl 1220.47018

[2] Birman, Mikhail; Solomyak, Michael Double Stieltjes operator integrals, Probl. Mat. Fiz. (1966), pp. 33-67 | Zbl 0161.34602

[3] Birman, Mikhail; Solomyak, Michael Spectral theory of selfadjoint operators in Hilbert space, Mathematics and its Applications, Soviet Series, Tome 5, Kluwer Academic Publishers, 1987, xvi+301 pages | Zbl 0744.47017

[4] Birman, Mikhail; Solomyak, Michael Operator integration, perturbations and commutators, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., Tome 170 (1989), pp. 34-66 | Zbl 0709.47022

[5] Birman, Mikhail; Solomyak, Michael Double operator integrals in a Hilbert space, Integral Equations Oper. Theory, Tome 47 (2003) no. 2, pp. 131-168 | Zbl 1054.47030

[6] Cadilhac, Léonard Weak boundedness of Calderón-Zygmund operators on noncommutative L $^{1}$ -spaces (2017) (https://arxiv.org/abs/1702.06536)

[7] Caspers, Martijn; Montgomery-Smith, Stephen; Potapov, Denis; Sukochev, Fedor The best constants for operator Lipschitz functions on Schatten classes, J. Funct. Anal., Tome 267 (2014) no. 10, pp. 3557-3579 | Zbl 06354029

[8] Caspers, Martijn; Potapov, Denis; Sukochev, Fedor; Zanin, Dmitriy Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture (to appear in Am. J. Math.)

[9] Caspers, Martijn; Potapov, Denis; Sukochev, Fedor; Zanin, Dmitriy Weak type estimates for the absolute value mapping, J. Oper. Theory, Tome 73 (2015) no. 2, pp. 361-384 | Zbl 06465646

[10] Connes, Alain; Marcolli, Matilde Noncommutative geometry, quantum fields and motives, Colloquium Publications, Tome 55, American Mathematical Society, 2008, xxii+785 pages | Zbl 1209.58007

[11] Davies, Edward Lipschitz continuity of functions of operators in the Schatten classes, J. Lond. Math. Soc., Tome 37 (1988), pp. 148-157 | Zbl 0648.47011

[12] Dodds, Peter; Dodds, Theresa; de Pagter, Ben; Sukochev, Fedor Lipschitz continuity of the absolute value and Riesz projections in symmetric operator spaces, J. Funct. Anal., Tome 148 (1997) no. 1, pp. 28-69 | Zbl 0899.46052

[13] Dodds, Peter; Dodds, Theresa; de Pagter, Ben; Sukochev, Fedor Lipschitz continuity of the absolute value in preduals of semifinite factors, Integral Equations Oper. Theory, Tome 34 (1999) no. 1, pp. 28-44 | Zbl 0946.46051

[14] Farforovskaya An example of a Lipschitz function of self-adjoint operators with non-nuclear difference under a nuclear perturbation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., Tome 30 (1972), pp. 146-153 | Zbl 0339.47012

[15] Grafakos, Loukas Classical Fourier analysis, Graduate Texts in Mathematics, Tome 249, Springer, 2014, xvii+638 pages | Zbl 1304.42001

[16] Hytönen, Tuomas; van Neerven, Jan; Veraar, Mark; Weis, Lutz Analysis in Banach spaces. Volume I: Martingales and Littlewood-Paley theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Tome 63, Springer, 2016, xvii+614 pages | Zbl 1366.46001

[17] Kalton, Nigel; Sukochev, Fedor Symmetric norms and spaces of operators, J. Reine Angew. Math., Tome 621 (2008), pp. 81-121 | Zbl 1152.47014

[18] Kato, Tosio Continuity of the map $S \mapsto | S |$ for linear operators, Proc. Japan Acad., Tome 49 (1973), pp. 157-160 | Zbl 0301.47006

[19] Kissin, Edward; Potapov, Denis; Shulman, Viktor; Sukochev, Fedor Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations, Proc. Lond. Math. Soc., Tome 105 (2012) no. 4, pp. 661-702 | Zbl 1258.47022

[20] Kosaki, Hideki Unitarily invariant norms under which the map $A \mapsto | A |$ is continuous, Publ. Res. Inst. Math. Sci., Tome 28 (1992) no. 2, pp. 299-313 | Zbl 0817.47013

[21] Kreĭn, Mark Some new studies in the theory of perturbations of self-adjoint operators., First mathematical summer school, part I (Kanev, 1963), Naukova Dumka, 1964, pp. 103-187 | Zbl 1245.47005

[22] de Leeuw, Karel On $L_{p}$ multipliers, Ann. Math., Tome 81 (1965), pp. 364-379 | Zbl 0171.11803

[23] Lord, Steven; Sukochev, Fedor; Zanin, Dmitriy Singular traces. Theory and applications, de Gruyter Studies in Mathematics, Tome 46, de Gruyter, 2013, xvi+452 pages | Zbl 1275.47002

[24] Nazarov, Fedor; Peller, Vladimir Lipschitz functions of perturbed operators, C. R. Math. Acad. Sci. Paris, Tome 347 (2009) no. 15-16, pp. 857-862 | Zbl 1169.47015

[25] de Pagter, Ben; Sukochev, Fedor Differentiation of operator functions in non-commutative $L_{p}$ -spaces, J. Funct. Anal., Tome 212 (2004) no. 1, pp. 28-75 | Zbl 1075.46054

[26] de Pagter, Ben; Sukochev, Fedor; Witvliet, H. Double operator integrals, J. Funct. Anal., Tome 192 (2002) no. 1, pp. 52-111 | Zbl 1079.47502

[27] Parcet, Javier Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory, J. Funct. Anal., Tome 256 (2009) no. 2, pp. 509-593 | Zbl 1179.46051

[28] Peller, Vladimir Hankel operators in the theory of perturbations of unitary and selfadjoint operators, Funkts. Anal. Prilozh., Tome 19 (1985) no. 2, pp. 37-51

[29] Pisier, Gilles Introduction to operator space theory, London Mathematical Society Lecture Note Series, Tome 294, Cambridge University Press, 2003, viii+478 pages | Zbl 1093.46001

[30] Potapov, Denis; Sukochev, Fedor Unbounded Fredholm modules and double operator integrals, J. Reine Angew. Math., Tome 626 (2009), pp. 159-185 | Zbl 1189.46063

[31] Potapov, Denis; Sukochev, Fedor Operator-Lipschitz functions in Schatten-von Neumann classes, Acta Math., Tome 207 (2011) no. 2, pp. 375-389 | Zbl 1242.47013

[32] Skripka, Anna Asymptotic expansions for trace functionals, J. Funct. Anal., Tome 266 (2014) no. 5, pp. 2845-2866 | Zbl 1317.47018

[33] Stein, Eliad Singular integrals and differentiability properties of functions, Princeton Mathematical Series, Tome 30 (1970), xiv+387 pages | Zbl 0207.13501

[34] van Suijlekom, Walter Perturbations and operator trace functions, J. Funct. Anal., Tome 260 (2011) no. 8, pp. 2483-2496 | Zbl 1218.46043

[35] Sukochev, Fedor Completeness of quasi-normed symmetric operator spaces, Indag. Math., Tome 25 (2014) no. 2, pp. 376-388 | Zbl 1298.46051

[36] Voiculescu, Dan Some results on norm-ideal perturbations of Hilbert space operators, J. Oper. Theory, Tome 2 (1979), pp. 3-37 | Zbl 0446.47003

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT