Schur multipliers of Schatten--von Neumann classes $\boldsymbol{S_p}$

TL;DR

This paper investigates Schur multipliers of Schatten classes, establishing their complete boundedness for p ≤ 1 and introducing a new tensor product scale that characterizes such multipliers.

Contribution

It proves that Schur multipliers of Schatten classes are completely bounded for p ≤ 1 and introduces the ${\\mathscr W}_p$ tensor product scale as a new characterization.

Findings

Schur multipliers of S_p are completely bounded for p ≤ 1.

Matrices in the new scale ${\mathscr W}_p$ are Schur multipliers of S_p.

Comparison with existing tensor product conditions is provided.

Abstract

We study in this paper properties of Schur multipliers of Schatten von Neumann classes $\boldsymbol{S}_p$. We prove that for $p\le1$, Schur multipliers of $\boldsymbol{S}_p$ are necessarily completely bounded. We also introduce for $p\le1$ a scale ${\mathscr W}_p$ of tensor products of $\ell^\infty$ and prove that matrices in ${\mathscr W}_p$ are Schur multipliers of $\boldsymbol{S}_p$. We compare this sufficient condition with the sufficient condition of membership in the $p$-tensor product of $\ell^\infty$ spaces.