, Volume 319, Issue 1, pp 1-16

Operator Khintchine inequality in non-commutative probability

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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.

Received November 6, 1998 / in final form June 8, 2000 / Published online December 8, 2000