On the complete bounds of $L_p$-Schur multipliers

Abstract

We study the class ${\mathcal{M}}_p$ of Schur multipliers on the Schatten-von Neumann class ${\mathcal{S}}_p$ with $1\le p\le \infty$ as well as the class of completely bounded Schur multipliers ${\mathcal{M}}_p^{cb}$. We first show that for $2\le p<q\le \infty$ there exist $m\in {\mathcal{M}}_p^{cb}$ with $m\notin {\mathcal{M}}_q$, so in particular the following inclusions that follow from interpolation are strict ${\mathcal{M}}_q⊊{\mathcal{M}}_p$ and ${\mathcal{M}}_q^{cb}⊊{\mathcal{M}}_p^{cb}$. In the remainder of the paper we collect computational evidence that for $p\ne 1,2,\infty$ we have ${\mathcal{M}}_p={\mathcal{M}}_p^{cb}$, moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier is false.

to appear in Archiv der Mathematik