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Complete boundedness of multiple operator integrals

Part of: Selfadjoint operator algebras Special classes of linear operators

Published online by Cambridge University Press:  27 July 2020

Clément Coine
Clément Coine*
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha410085, People’s Republic of China
*
e-mail: clement.coine1@gmail.com
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Abstract

In this paper, we characterize the multiple operator integrals mappings that are bounded on the Haagerup tensor product of spaces of compact operators. We show that such maps are automatically completely bounded and prove that this is equivalent to a certain factorization property of the symbol associated with the operator integral mapping. This generalizes a result by Juschenko-Todorov-Turowska on the boundedness of measurable multilinear Schur multipliers.

Keywords

Complete boundednessmultiplieroperator integrals

MSC classification

Primary: 46L07: Operator spaces and completely bounded maps 47B49: Transformers, preservers (operators on spaces of operators)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 474 - 490
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The author is supported by NSFC(11801573).

References

Birman, M. and Solomyak, M., Double Stieltjes operator integrals . Prob. Math. Phys. Izdat. Leningrad Univ. 1(1966), 33–67 [in Russian]. https://doi.org/10.1007/978-1-4684-7595-1_2 Google Scholar
Birman, M. and Solomyak, M., Double Stieltjes operator integrals II. Prob. Math. Phys. Izdat. Leningrad Univ. 2(1967), 26–60 [in Russian]. https://doi.org/10.1007/978-1-4684-7592-0_3Google Scholar
Birman, M. and Solomyak, M., Double Stieltjes operator integrals III . Prob. Math. Phys. Izdat. Leningrad Univ. 6(1973), 27–53 [in Russian].Google Scholar
Blecher, D.P. and Le Merdy, C., Operator algebras and their modules – an operator space approach . Oxford University Press, Oxford, UK, 2004.CrossRefGoogle Scholar
Coine, C., Le Merdy, C., and Sukochev, F., When do triple operator integrals take value in the trace class? Preprint, 2017. arXiv:1706.01662 Google Scholar
Coine, C., Le Merdy, C., Sukochev, F., and Skripka, A., Higher order ${\boldsymbol{\mathcal{S}}}^2$ -differentiability and application to Koplienko trace formula. J. Funct. Anal., to appear. https://doi.org/10.1016/j.jfa.2018.09.00 CrossRefGoogle Scholar
Coine, C., Perturbation theory and higher order ${\boldsymbol{\mathcal{S}}}^p$ -differentiability of operator functions. Preprint, 2019. arXiv:1906.05585 Google Scholar
Conway, J., A course in operator theory . Graduate Studies in Mathematics, Vol. 21, American Mathematical Society, Providence, RI, 2000.Google Scholar
Diestel, J. and Uhl, J. J., Vector measures . Mathematical Surveys, 15, American Mathematical Society, Providence, RI, 1979.Google Scholar
Dunford, N. and Pettis, B., Linear operator on summable functions . Trans. Amer. Math. Soc. 47(1940), 323–392. https://doi.org/10.1090/s0002-9947-1940-0002020-4 CrossRefGoogle Scholar
Effros, E. G. and Ruan, Zh.-J., Multivariable multipliers for groups and their operator algebras. In: Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), Proc. Sympos. Pure Math., 51, Part 1, Amer. Math. Soc., Providence, RI, 1990, pp. 197–218.Google Scholar
Caspers, M. and Wildschut, G., On the complete bounds of ${L}_p$ -Schur multipliers. Arch. Math. 113(2019), 189. https://doi.org/10.1007/s00013-019-01316-7 CrossRefGoogle Scholar
Juschenko, K., Todorov, I. G., and Turowska, L., Multidimensional operator multipliers . Trans. Amer. Math. Soc. 361(2009), no. 9, 4683–4720. https://doi.org/10.1090/s0002-9947-09-04771-0 CrossRefGoogle Scholar
Le Merdy, C. and Skripka, A., Higher order differentiability of operator functions in Schatten norms. J. Inst. Math. Jussieu 19 (2020), 1993–2016. https://doi.org/10.1017/s1474748019000033 CrossRefGoogle Scholar
Pavlov, B., Multidimensional operator integrals. Problems of Math. Anal., No. 2, Linear Operators and Operator Equations (1969), 99–122 [in Russian].Google Scholar
Peller, V. V., Multiple operator integrals and higher operator derivatives . J. Funct. Anal. 233(2006), no. 2, 515–544. https://doi.org/10.1016/j.jfa.2005.09.003 CrossRefGoogle Scholar
Pisier, G., Introduction to operator space theory. London Mathematical Society, Lecture note Series, 294, London, UK, 2003.CrossRefGoogle Scholar
Pisier, G., Similarity problems and completely bounded maps. Lecture Notes in Mathematics, 1618, Springer-Verlag, Berlin, 1996.CrossRefGoogle Scholar
Potapov, D., Skripka, A., and Sukochev, F., Spectral shift function of higher order. Invent. Math. 193(2013), no. 3, 501–538. https://doi.org/10.1007/s00222-012-0431-2 CrossRefGoogle Scholar
Effros, E. G. and Ruan, Z., Operator spaces. London Mathematical Society Monographs New Series, 23, London, UK, 2000.Google Scholar
Spronk, N., Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras. Proc. Lond. Math. Soc. (3) 89(2004), 161–192. https://doi.org/10.1112/s0024611504014650 CrossRefGoogle Scholar
Stenkin, V. V., Multiple operator integrals. Izv. Vysh. Uchebn. Zaved. Matematika 4(1977), 102–115 [in Russian]. English transl.: Soviet Math. (Iz. VUZ) 21(1977), no. 4, 88–99.Google Scholar