Operator Khintchine inequality in non-commutative probability

Abstract. For the operator $T=\sum^N_{k=1}A_K \otimes R_k$, where $A_k$ belongs to the Schatten class $S_{2n}$ and where $R_k$ are non-commutative random variables with mixed moments satisfying a specific condition, we prove the following Khintchine inequality

\[ \Vert{T}\Vert_{2n} \leq D_{2n} \left \{ \left \Vert \left (\sum^N_{k001}A_kA_k^* \right )^{\frac{1}{2} \right \Vert_{S_{2n}}\quad ,\quad \left \Vert \left ( \sum^N_{k=1}A_K^*A_k\right )^{\frac{1}{2} \right \Vert_{S_{2n}}\right \}. \]

We find the optimal constants $D_{2n}$ in the case when $R_{k}$ are the q-Gaussian and circular random variables. Moreover, we show that the moments of any probability symmetric measure appear as the optimal constants for some random variables.