Journal of Operator Theory

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

Schur multipliers on $\mathcal{B}(L^p,L^q)$

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

Summary: Let $(\Omega_1, \mathcal{F}_1, \mu_1)$ and $(\Omega_2, \mathcal{F}_2, \mu_2)$ 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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