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
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