Weak type operator Lipschitz and commutator estimates for commuting tuples

Caspers, Martijn; Sukochev, Fedor; Zanin, Dmitriy

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

Soit $f:{ℝ}^d\to ℝ$ une fonction Lipschitzienne. Si $B$ est un opérateur borné auto-adjoint et si ${\left\{A_k\right\}}_{k=1}^d$ sont des opérateurs bornés auto-adjoints qui commutent et tels que $[A_k,B]\in L_1\left(H\right),$ alors

$$\parallel [f(A_1,\cdots ,A_d),B]{\parallel }_{1,\infty }\le c\left(d\right)\parallel \nabla \left(f\right){\parallel }_{\infty }\underset{1\le k\le d}{max}{\parallel [A_k,B]\parallel }_1,$$

où $c\left(d\right)$ est une constante indépendante de $f$, $ℳ$ et $A,B$ et ${\parallel ·\parallel }_{1,\infty }$ désigne la norme $L_1$-faible.

Si ${\left\{X_k\right\}}_{k=1}^d$ (respectivement ${\left\{Y_k\right\}}_{k=1}^d$) sont des opérateurs bornés qui commutent et tels que $X_k-Y_k\in L_1\left(H\right),$ alors

$$\parallel f(X_1,\cdots ,X_d)-f(Y_1,\cdots ,Y_d){\parallel }_{1,\infty }\le c\left(d\right)\parallel \nabla \left(f\right){\parallel }_{\infty }\underset{1\le k\le d}{max}{\parallel X_k-Y_k\parallel }_1.$$

Let $f:{ℝ}^d\to ℝ$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if ${\left\{A_k\right\}}_{k=1}^d$ are commuting bounded self-adjoint operators such that $[A_k,B]\in L_1\left(H\right),$ then

$$\parallel [f(A_1,\cdots ,A_d),B]{\parallel }_{1,\infty }\le c\left(d\right)\parallel \nabla \left(f\right){\parallel }_{\infty }\underset{1\le k\le d}{max}{\parallel [A_k,B]\parallel }_1,$$

where $c\left(d\right)$ is a constant independent of $f$, $ℳ$ and $A,B$ and ${\parallel ·\parallel }_{1,\infty }$ denotes the weak $L_1$-norm.

If ${\left\{X_k\right\}}_{k=1}^d$ (respectively, ${\left\{Y_k\right\}}_{k=1}^d$) are commuting bounded self-adjoint operators such that $X_k-Y_k\in L_1\left(H\right),$ then

$$\parallel f(X_1,\cdots ,X_d)-f(Y_1,\cdots ,Y_d){\parallel }_{1,\infty }\le c\left(d\right)\parallel \nabla \left(f\right){\parallel }_{\infty }\underset{1\le k\le d}{max}{\parallel X_k-Y_k\parallel }_1.$$

Classification : 47B10, 47L20, 47A30
Keywords: Non-commutative $L_p$-spaces, commutator estimates, Calderón–Zygmund theory
Mot clés : Espaces $L_p$ non commutatifs, estimées de commutateurs, théorie de Calderón–Zygmund

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

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

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

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

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

[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., Volume 267 (2014) no. 10, pp. 3557-3579 | Zbl

[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, Volume 73 (2015) no. 2, pp. 361-384 | Zbl

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

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

[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., Volume 148 (1997) no. 1, pp. 28-69 | Zbl

[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, Volume 34 (1999) no. 1, pp. 28-44 | Zbl

[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., Volume 30 (1972), pp. 146-153 | Zbl

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

[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, 63, Springer, 2016, xvii+614 pages | Zbl

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

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

[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., Volume 105 (2012) no. 4, pp. 661-702 | Zbl

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

[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

[22] de Leeuw, Karel On $L_p$ multipliers, Ann. Math., Volume 81 (1965), pp. 364-379 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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