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5. Schur multipliers and Grothendieck’s inequality

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

Abstract

In this chapter, we study Schur multipliers on the space B(H, K) of all bounded operators between two Hilbert spaces. We give a basic characterization of the unit ball of the space of Schur multipliers, in connection with the class of operators factoring through a Hilbert space (considered above in chapter 3). Then we prove Grothendieck’s fundamental theorem (= Grothendieck’s inequality) in terms of Schur multipliers. We give Varopoulos’s proof that, since the Grothendieck constant is > 1, Ando’s inequality does not extend with constant 1 to n-tuples of mutually commuting contractions. Finally, we discuss the extensions to Schur multipliers acting boundedly on the space B(H, K) when H, K are replaced by \(\ell_p\)-spaces, \(1 \leq p < \infty\).

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). 5. Schur multipliers and Grothendieck’s inequality. 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_6

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  • DOI: https://doi.org/10.1007/978-3-540-44563-0_6

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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