A special case of the existential version of the Non Commutative Khintchine inequality

Abstract

Here we prove the following result. Let $A = \{a_{ij}\}_{i,j\in \mathbb{N}}$ be a bounded operator. Then there exists a signing of $A$ such that $$||A\circ S||_2 < 2||A||_{l_\infty(l_2)},$$ where $A\circ S$ denotes the matrix generated by the entry-wise product of $A$ and $S$.