Remarks on the non-commutative Khintchine inequalities for 0<p<2

AbstractWe show that the validity of the non-commutative Khintchine inequality for some q with 1<q<2 implies its validity (with another constant) for all 1⩽p<q. We prove this for the inequality involving the Rademacher functions, but also for more general “lacunary” sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the “Z(2)-sequences” previously considered by Harcharras. The result appears to be new in that case. It implies that the space ℓ1n contains (as an operator space) a large subspace uniformly isomorphic (as an operator space) to Rk+Ck with k∼n12. This naturally raises several interesting questions concerning the best possible such k. Unfortunately we cannot settle the validity of the non-commutative Khintchine inequality for 0<p<1 but we can prove several would be corollaries. For instance, given an infinite scalar matrix [xij], we give a necessary and sufficient condition for [±xij] to be in the Schatten class Sp for almost all (independent) choices of signs ±1. We also characterize the bounded Schur multipliers from S2 to Sp. The latter two characterizations extend to 0<p