Sharp Operator Khintchine inequality and $Z_2$ Property

Abstract

We study the $p=1$ operator Khintchine inequality associated with orthonormal systems and establish two quantitative implications relating its optimal constant $A_1$ to the $Z_2$ property. For an orthonormal system $W$ with finite $Z_2(W)$, we prove $A_1(W)\le \sqrt{1+Z_2(W)}$. Conversely, there exists an absolute constant $\delta >0$ such that, for canonical group unitaries indexed by any subset $V$ of a discrete group, $A_1(V)<\sqrt{2}+\delta$ implies $Z_2(V)\le 2$. The converse is detected by $2\times 2$ matrix coefficients supported on at most six group elements. For the classical Rademacher sequence, we determine the sharp operator Khintchine constant $A_1=\sqrt{2}$, answering the question left open by Haagerup and Musat~\cite{Haagerup2007}. Since this sequence has $Z_2=2$, the conclusion $Z_2(V)\le 2$ is optimal.