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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© 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
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