Skip to main content
×
×
Home

Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group

  • Stefan Neuwirth (a1) and Éric Ricard (a1)
Abstract

We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue– Orlicz spaces of a discrete group and relative Toeplitz-Schur multipliers on Schatten–von- Neumann–Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum , the norm of the Hilbert transformand the Riesz projection on Schatten–von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten–von-Neumann classes with exponent less than 1.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group
      Available formats
      ×
Copyright
References
Hide All
[1] Aleksandrov, A. B and Peller, V. V., Hankel and Toeplitz-Schur multipliers. Math. Ann. 324(2002), no. 2, 277–327. doi:10.1007/s00208-002-0339-z
[2] Bédos, E., On Følner nets, Szegő’s theorem and other eigenvalue distribution theorems. Exposition. Math. 15(1997), no. 3, 193–228, 384.
[3] Bourgain, J., Λp-sets in analysis: results, problems and related aspects. In: Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 195–232. p
[4] Bozejko, M., A new group algebra and lacunary sets in discrete noncommutative groups. Studia Math. 70(1981), no. 2, 165–175.
[5] Bozejko, M. and Fendler, G., Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group. Boll. Un. Mat. Ital. A (6) 3(1984), no. 2, 297–302.
[6] Brown, N.P. and Ozawa, N., C -algebras and finite-dimensional approximations. Graduate Studies in Mathematics, 88, American Mathematical Society, Providence, RI, 2008.
[7] Cannière, J.De. and Haagerup, U., Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107(1985), no. 2, 455–500. doi:10.2307/2374423
[8] Gamelin, T.W., Uniform algebras and Jensen measures. London Mathematical Society Lecture Note Series, 32, Cambridge University Press, Cambridge-New York, 1978.
[9] Gohberg, I. and Krupnik, N., One-dimensional linear singular integral equations. I. Operator Theory: Advances and Applications, 53, Birkhäuser Verlag, Basel, 1992.
[10] Gokhberg, I.Ts. and Krupnik, N. Ya., Norm of the Hilbert transformation in the L space. Russian, Funktsional. Anal. i Prilozhen. 2(1968), no. 2, 91–92; English translation in Funct. Anal. Appl. 2(1968), no. 2, 180–181. doi:10.1007/BF01075955
[11] Haagerup, U. and Kraus, J., Approximation properties for group C -algebras and group von Neumann algebras. Trans. Amer. Math. Soc. 344(1994), no. 2, 667–699. doi:10.2307/2154501
[12] Harcharras, A., Fourier analysis, Schur multipliers on Sp and non-commutative Λ(p)-sets. Studia Math. 137(1999), no. 3, 203–260.
[13] Hollenbeck, B., N. J. Kalton, and Verbitsky, I. E., Best constants for some operators associated with the Fourier and Hilbert transforms. Studia Math. 157(2003), no. 3, 237–278. doi:10.4064/sm157-3-2 p
[14] Hollenbeck, B. and Verbitsky, I. E., Best constants for the Riesz projection. J. Funct. Anal. 175(2000), no. 2, 370–392.
[15] Junge, M. and Ruan, Z.-J., Approximation properties for noncommutative L -spaces associated with discrete groups. Duke Math. J. 117(2003), no. 2, 313–341. doi:10.1215/S0012-7094-03-11724-X (T) spaces. Studia Math. 121(1996), no. 3, 231–247.
[16] Li, D., Complex unconditional metric approximation property for C Λ p
[17] Li, D. and Queffélec, H., Introduction à l’étude des espaces de Banach. Analyse et probabilités. Cours Spécialisés, 12, Société Mathématique de France, Paris, 2004.
[18] López, J.M. and Ross, K. A., Sidon sets. Lecture Notes Pure and Applied Mathematics, 13, Marcel Dekker Inc., New York, 1975.
[19] Marsalli, M. and West, G., Noncommutative H p spaces. J. Operator Theory 40(1998), no. 2, 339–355, 1998.
[20] Nazarov, F., Pisier, G., Treil, S., and Volberg, A., Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts. J. Reine Angew. Math. 542(2002), 147–171. doi:10.1515/crll.2002.004
[21] Neuwirth, S., Multiplicateurs et analyse fonctionnelle. Ph.D. thesis, Université Paris 6, 1999. http://tel.archives-ouvertes.fr/tel-00010399
[22] Neuwirth, S., Cycles and 1-unconditional matrices. Proc. London Math. Soc. (3) 93(2006), no. 3, 761–790. doi:10.1017/S0024611506015899
[23] Oberlin, D.M., Translation-invariant operators on L p (G), 0 < p < 1. Michigan Math. J. 23(1976), no. 2, 119–122.
[24] Oikhberg, T., Restricted Schur multipliers and their applications. Proc. Amer. Math. Soc. 138(2010), no. 5, 1739–1750. doi:10.1090/S0002-9939-10-10203-2
[25] Olevskii, V., A connection between Fourier and Schur multipliers. Integral Equations Operator Theory 25(1996), no. 4, 496–500. doi:10.1007/BF01203030
[26] Orlicz, W., Linear functional analysis. Series in Real Analysis, 4. World Scientific Publishing Co., Inc., River Edge, NJ, 1992.
[27] Paulsen, V., Completely bounded maps and operator algebras. Cambridge Studies in Advanced Mathematics, 78, Cambridge University Press, Cambridge, 2002.
[28] Paulsen, V.I., Power, S. C., and Smith, Roger R., Schur products and matrix completions. J. Funct. Anal. 85(1989), 151–178. doi:10.1016/0022-1236(89)90050-5
[29] Peller, V.V., Hankel operators and their applications. Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.
[30] Pichorides, S.K.. On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov. Studia Math. 44(1972), 165–179 (errata insert). spaces and completely p-summing maps. Astérisque 247(1998), 131 pp.
[31] Pisier, G., Non-commutative vector valued L p
[32] Pisier, G., Similarity problems and completely bounded maps. Second, expanded ed., includes the solution to “The Halmos problem”, Lecture Notes in Mathematics, 1618, Springer-Verlag, Berlin, 2001.
[33] Randrianantoanina, N., Hilbert transform associated with finite maximal subdiagonal algebras. J. Austral. Math. Soc. Ser. A 65(1998), no. 3, 388–404.
[34] Ricard, É., L’espace H 1 n’a pas de base complètement inconditionnelle. C. R. Acad. Sci. Paris Sér. I Math. 331(2000), no. 8, 625–628. doi:10.1016/S0764-4442(00)01680-3
[35] Ricard, É., Décompositions de H 1 , multiplicateurs de Schur et espaces d’opérateurs. Ph.D. thesis, Université Paris 6, 2001. http://www.institut.math.jussieu.fr/theses/2001/ricard
[36] Rudin, W., Trigonometric series with gaps. J. Math. Mech. 9(1960), no. 2, 203–227.
[37] Varopoulos, N.Th., Tensor algebras over discrete spaces. J. Functional Analysis 3(1969), no. 2, 321–335. doi:10.1016/0022-1236(69)90046-9
[38] Zsid, L.ó, On spectral subspaces associated to locally compact abelian groups of operators. Adv. in Math. 36, no. 3, 213–276. doi:10.1016/0001-8708(80)90016-X
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed