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

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 $$\mathrm{\mid }\mathrm{\mid }A\circ S\mathrm{\mid }{\mathrm{\mid }}_2<2\mathrm{\mid }\mathrm{\mid }A\mathrm{\mid }{\mathrm{\mid }}_{l_{\mathrm{\infty }}(l_2)},$$ where $A\circ S$ denotes the matrix generated by the entry-wise product of $A$ and $S$.