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On a conjecture of Pisier on the analyticity of semigroups

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Abstract

We show that the analyticity of semigroups \((T_t)_{t \geqslant 0}\) of selfadjoint contractive Fourier multipliers on \(L^p\)-spaces of compact abelian groups is preserved by the tensorisation of the identity operator of a Banach space for a large class of K-convex Banach spaces, answering partially a conjecture of Pisier. We also give versions of this result for some semigroups of Schur multipliers and Fourier multipliers on noncommutative \(L^p\)-spaces. Finally, we give a precise description of semigroups of Schur multipliers to which the result of this paper can be applied.

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References

  1. Arhancet, C.: On Matsaev’s conjecture for contractions on noncommutative \(L^p\)-spaces. J. Oper. Theory 69(2), 387–421 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arhancet, C.: Analytic semigroups on vector valued noncommutative \(L^p\)-spaces. Stud. Math. 216(3), 271–290 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arhancet, C.: Semigroups of operators and OK-convexity (in preparation)

  4. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Monographs in Mathematics. Vector-valued Laplace transforms and Cauchy problems, vol. 96, 2nd edn. Birkhäuser, Basel (2011)

    Google Scholar 

  5. Berg, C., Christensen, J., Ressel, P.: Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions. Springer, New York (1984)

    Book  MATH  Google Scholar 

  6. Bekka, B., de la Harpe, P., Valette, A.: New Mathematical Monographs. Kazhdan’s property (T), vol. 11. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  7. Beurling, A.: On analytic extension of semigroups of operators. J. Funct. Anal. 6, 387–400 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  8. De Cannière, J., Haagerup, U.: Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Am. J. Math. 107(2), 455–500 (1985)

    Article  MATH  Google Scholar 

  9. Diestel, J., Jarchow, H., Tonge, A.: Cambridge Studies in Advanced Mathematics. Absolutely summing operators, vol. 43. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  10. Engel, K.-J., Nagel, R.: Graduate Texts in Mathematics. One-parameter semigroups for linear evolution equations, vol. 194. Springer, New York (2000)

    Google Scholar 

  11. Effros, E., Ruan, Z.-J.: Operator Spaces. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  12. Fackler, S.: Regularity of semigroups via the asymptotic behaviour at zero. Semigroup Forum 87(1), 117 (2013)

    Article  MathSciNet  Google Scholar 

  13. Haagerup, U., Musat, M.: Factorization and dilation problems for completely positive maps on von Neumann algebras. Commun. Math. Phys. 303(2), 555–594 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Haase, M.: Operator Theory: Advances and Applications. The functional calculus for sectorial operators, vol. 169. Birkhäuser, Basel (2006)

    Google Scholar 

  15. Hinrichs, A.: \(K\)-convex operators and Walsh type norms. Math. Nachr. 208, 121–140 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Knudby, S.: Semigroups of Herz-Schur multipliers. J. Funct. Anal. 266(3), 1565–1610 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Junge, M., Le Merdy, C., Xu, Q.: \(H^\infty \) functional calculus and square functions on noncommutative \(L^p\)-spaces. Astérisque 305, vi+138 (2006)

    Google Scholar 

  18. Junge, M., Parcet, J.: The norm of sums of independent noncommutative random variables in \(L_p(l_1)\). J. Funct. Anal. 221(2), 366–406 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Losert, V.: Properties of the Fourier algebra that are equivalent to amenability. Proc. Am. Math. Soc. 92(3), 347–354 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  20. Neuwirth, S., Ricard, É.: Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group. Can. J. Math. 63(5), 1161–1187 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  22. Pisier, G.: Semi-groupes holomorphes et \(K\)-convexité. (French) [Holomorphic semigroups and \(K\)-convexity]. Seminar on Functional Analysis, 1980–1981, Exp. No. II, p. 32, École Polytechnique, Palaiseau (1981)

  23. Pisier, G.: Semi-groupes holomorphes et \(K\)-convexité (suite). (French) [Holomorphic semigroups and \(K\)-convexity (continued)] Seminar on Functional Analysis, 1980–1981, Exp. No. VII, p. 10, École Polytechnique, Palaiseau (1981)

  24. Pisier, G.: Holomorphic semigroups and the geometry of Banach spaces. Ann. Math. 115(2), 375–392 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pisier, G.: Regular operators between non-commutative \(L_p\)-spaces. Bull. Sci. Math. 119(2), 95–118 (1995)

    MATH  MathSciNet  Google Scholar 

  26. Pisier, G.: Non-commutative vector valued \(L_p\)-spaces and completely \(p\)-summing maps. Astérisque 247, vi+131 (1998)

    MathSciNet  Google Scholar 

  27. Pisier, G.: Similarity Problems and Completely Bounded Maps. Lecture notes in mathematics, vol. 1618. Springer, New York (2001)

    Book  MATH  Google Scholar 

  28. Pisier, G.: Introduction to Operator Space Theory. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  29. Pisier, G.: Remarks on the non-commutative Khintchine inequalities for 0\(<\)p\(<\)2. J. Funct. Anal. 256(12), 4128–4161 (2009)

  30. Pisier, G., Xu, Q.: Non-commutative \(L^p\)-spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517. Elsevier, Amsterdam (2003)

    Chapter  Google Scholar 

  31. Ricard, É.: A Markov dilation for self-adjoint Schur multipliers. Proc. Am. Math. Soc. 136(12), 4365–4372 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sinclair, A., Smith, R.: London Mathematical Society Lecture Note Series. Finite von Neumann algebras and masas, vol. 351. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  33. Spronk, N.: Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras. Proc. London Math. Soc. 89(1), 161–192 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Stein, E.M.: Annals of Mathematics Studies. Topics in harmonic analysis related to the Littlewood-Paley theory, vol. 63. Princeton University Press/University of Tokyo Press, Princeton/Tokyo (1970)

    Google Scholar 

  35. Stratila, S.: Modular Theory in Operator Algebras. Taylor and Francis, Florence (1981)

    MATH  Google Scholar 

  36. Tao, T.: Ultraproducts as a Bridge Between Discrete and Continuous Analysis. https://terrytao.wordpress.com/2013/12/07/ultraproducts-as-a-bridge-between-discrete-and-continuous-analysis/

  37. Xu, Q.: \(H^\infty \) functional calculus and maximal inequalities for semigroups of contractions on vector-valued \(L_p\)-spaces (in press), arXiv:1402.2344

Download references

Acknowledgments

The author would like to thank to Christian Le Merdy to provide him with the preprint [37] and Marius Junge and Eric Ricard for some discussions. Finally, the referee deserves thanks for a careful reading of the paper. The author is supported by the research program ANR 2011 BS01 008 01.

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  1. Laboratoire de Mathématiques, Université de Franche-Comté, 25030, Besançon Cedex, France

    Cédric Arhancet

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  1. Cédric Arhancet
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Correspondence to Cédric Arhancet.

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Communicated by Markus Haase.

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Arhancet, C. On a conjecture of Pisier on the analyticity of semigroups. Semigroup Forum 91, 450–462 (2015). https://doi.org/10.1007/s00233-015-9715-3

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  • DOI: https://doi.org/10.1007/s00233-015-9715-3

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