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 $S$.