5. Schur multipliers and Grothendieck’s inequality

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, $1 \leq p < \infty$.