Operator Khintchine inequality in non-commutative probability

For the operator T = [GRAPHICS] A(k) x 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 [GRAPHICS] 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.