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Schur multipliers and Grothendieck’s inequality

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

Summary

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 l p -spaces, 1≤p < ∞.

Keywords

  • Hilbert Space
  • Banach Space
  • Unit Ball
  • Measure Space
  • Homogeneous Polynomial

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© 1996 Springer-Verlag Berlin Heidelberg

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Pisier, G. (1996). 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/10.1007/978-3-662-21537-1_6

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  • DOI: https://doi.org/10.1007/978-3-662-21537-1_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60322-1

  • Online ISBN: 978-3-662-21537-1

  • eBook Packages: Springer Book Archive