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Journal of Operator Theory

Volume 79, Issue 2, Spring 2018  pp. 301-326.

Schur multipliers on B(Lp,Lq)

Authors:  Clement Coine
Author institution: Laboratoire de Mathematiques de Besancon, UMR 6623, CNRS, Universite Bourgogne Franche-Comte, Besancon, 25000, France

Summary:  Let (Ω1,F1,μ1) and (Ω2,F2,μ2) be two measure spaces and 1 \leqslant p,q \leqslant +\infty. We give a definition of Schur multipliers on \mathcal{B}(L^p(\Omega_1), L^q(\Omega_2)) which extends the definition of classical Schur multipliers on \mathcal{B}(\ell_p,\ell_q). Our main result is a characterization of Schur multipliers in the case 1\leqslant q \leqslant p \leqslant +\infty. When 1 < q \leqslant p < +\infty, \phi \in L^{\infty}(\Omega_1 \times \Omega_2) is a Schur multiplier on \mathcal{B}(L^p(\Omega_1), L^q(\Omega_2)) if and only if there are a measure space (a probability space when p\neq q) (\Omega,\mu), a\in L^{\infty}(\mu_1, L^{p}(\mu)) and b\in L^{\infty}(\mu_2, L^{q'}(\mu)) such that, for almost every (s,t) \in \Omega_1 \times \Omega_2, \begin{equation*} \phi(s,t)= \langle a(s), b(t) \rangle. \end{equation*} This result is new, even in the classical case. As a consequence, we give new inclusion relationships among the spaces of Schur multipliers on \mathcal{B}(\ell_p,\ell_q).

DOI: http://dx.doi.org/10.7900/jot.2017mar23.2153
Keywords: multiplier, tensor product

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