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

TL;DR

This paper proves a specific case of an inequality involving non-commutative Khintchine inequalities, showing that for a bounded operator, a certain signed version has a controlled Hilbert-Schmidt norm.

Contribution

It establishes a particular instance of the existential non-commutative Khintchine inequality with explicit bounds for signed operators.

Findings

Existence of a signing with controlled norm

Bounded operator case analyzed

Inequality with explicit constant proved

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 $S$.