Comptes Rendus
Mathematical analysis/Functional analysis
Almost commuting functions of almost commuting self-adjoint operators
Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 583-588.

Let A and B be almost commuting (i.e, ABBAS1) self-adjoint operators. We construct a functional calculus φφ(A,B) for φ in the Besov class B,11(R2). This functional calculus is linear, the operators φ(A,B) and ψ(A,B) almost commute for φ,ψB,11(R2), φ(A,B)=u(A)v(B) whenever φ(s,t)=u(s)v(t), and the Helton–Howe trace formula holds. The main tool is triple operator integrals.

On dit que des opérateurs A et B sont presque commutants si leur commutateur [A,B] appartient à la classe trace. Pour des opérateurs A et B auto-adjoints qui presque commutent, nous construisons un calcul fonctionnel φφ(A,B), φB,11(R2), où B,11(R2) est la classe de Besov. Ce calcul a les propriétés suivantes : il est linéaire, les opérateurs φ(A,B) et ψ(A,B) presque commutent pour toutes les fonctions φ et ψ dans B,11(R2), φ(A,B)=u(A)v(B) si φ(s,t)=u(s)v(t), et la formule des traces de Helton et Howe est vraie. L'outil principal est la notion d'intégrales triples opératorielles.

Published online:
DOI: 10.1016/j.crma.2015.04.012

Aleksei Aleksandrov 1, 2; Vladimir Peller 3

1 St.-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St. Petersburg, Russia
2 Department of Mathematics and Mechanics, Saint Petersburg State University, 28, Universitetski pr., St. Petersburg, 198504, Russia
3 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
     author = {Aleksei Aleksandrov and Vladimir Peller},
     title = {Almost commuting functions of almost commuting self-adjoint operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {583--588},
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     year = {2015},
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Aleksei Aleksandrov; Vladimir Peller. Almost commuting functions of almost commuting self-adjoint operators. Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 583-588. doi : 10.1016/j.crma.2015.04.012.

[1] A.B. Aleksandrov; F.L. Nazarov; V.V. Peller Functions of perturbed noncommuting self-adjoint operators, C. R. Acad. Sci. Paris, Ser. I (2015) (in press)

[2] A.B. Aleksandrov; V.V. Peller Operator Hölder–Zygmund functions, Adv. Math., Volume 224 (2010), pp. 910-966

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

[4] R.W. Carey; J.D. Pincus Mosaics, principal functions and mean motion in von Neumann algebras, Acta Math., Volume 138 (1977), pp. 153-218

[5] K.F. Clancey Seminormal Operators, Lecture Notes in Mathematics, vol. 742, Springer-Verlag, Berlin, 1979

[6] J.W. Helton; R. Howe Integral operators, commutators, traces, index, and homology, Lecture Notes in Mathematics, vol. 345, Springer-Verlag, New York, 1973, pp. 141-209

[7] K. Juschenko; I.G. Todorov; L. Turowska Multidimensional operator multipliers, Trans. Amer. Math. Soc., Volume 361 (2009), pp. 4683-4720

[8] J. Peetre New Thoughts on Besov Spaces, Duke Univ. Press, Durham, NC, 1976

[9] V.V. Peller Hankel operators of class Sp and their applications (rational approximation, Gaussian processes, the problem of majorizing operators), Mat. Sb., Volume 113 (1980), pp. 538-581 English translation in Math. USSR Sbornik 41 (1982) 443–479

[10] V.V. Peller Hankel operators in the theory of perturbations of unitary and self-adjoint operators, Funkc. Anal. Prilozh., Volume 19 (1985) no. 2, pp. 37-51 (Russian). English translation: Funct. Anal. Appl. 19 (1985) 111–123

[11] V.V. Peller Hankel operators in the perturbation theory of unbounded self-adjoint operators, Analysis and Partial Differential Equations, Lecture Notes in Pure and Applied Mathematics, vol. 122, Dekker, New York, 1990, pp. 529-544

[12] V.V. Peller Functional calculus for a pair of almost commuting self-adjoint operators, J. Funct. Anal., Volume 112 (1993), pp. 325-345

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

[14] V.V. Peller Multiple operator integrals and higher operator derivatives, J. Funct. Anal., Volume 233 (2006), pp. 515-544

[15] J.D. Pincus Commutators and systems of singular integral equations, I, Acta Math., Volume 121 (1968), pp. 219-249

[16] J.D. Pincus, On the trace of commutators in the algebra of operators generated by an operator with trace class self-commutator, Stony Brook, preprint, 1972.

[17] G. Pisier Introduction to Operator Space Theory, London Math. Society Lect. Notes Series, vol. 294, Cambridge University Press, Cambridge, UK, 2003

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