An inequality for p-orthogonal sums in non-commutative ${\mathbf{L}}_{\mathbf{p}}$

We give an alternate proof of one of the inequalities proved recently for martingales (=sums of martingale differences) in a non-commutative $L_p$-space, with $1<p<\mathrm{\infty }$, by Q. Xu and the author. This new approach is restricted to $p$ an even integer, but it yields a constant which is $O(p)$ when $p\to \infty$anditappliestoamuchmoregeneralkindofsumswhichwecall$p$-orthogonal.