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6. Hankelian Schur multipliers. Herz-Schur multipliers

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1618)

Abstract

In this short chapter, we discuss Schur multipliers restricted to various subspaces \(E \subset B(H)\). We first discuss the case when \(H = \ell_2\) and E is the sub-class of all Hankel matrices. We show that the Schur multipliers which are completely bounded maps from E to E are closely related to the Fourier multipliers on the Hardy space H1. Analogously, when \(H = \ell_2(G)\) and E is the reduced C*-algebra \(C_{\lambda}^{*}(G)\), then the Schur multipliers which are completely bounded maps from E to E are identical to the completely bounded multipliers of \(C_{\lambda}^{*}(G)\) or equivalently to the (so-called) Herz-Schur multipliers of G.

Mathematics Subject Classification (2000):

  • primary 47AO5
  • 46LO5
  • 43A65 secondary 47A20
  • 47B 10
  • 42B30
  • 46E40
  • 46L57
  • 47C 15
  • 47L20

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Correspondence to Gilles Pisier .

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© 2001 Springer-Verlag Berlin/Heidelberg

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Pisier, G. (2001). 6. Hankelian Schur multipliers. Herz-Schur multipliers. In: Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics, vol 1618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44563-0_7

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  • DOI: https://doi.org/10.1007/978-3-540-44563-0_7

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41524-4

  • Online ISBN: 978-3-540-44563-0

  • eBook Packages: Springer Book Archive