An Interpolation Theorem for Quasimartingales in Noncommutative Symmetric Spaces

Let E be a separable symmetric space on (0, ∞) and E(M) the corresponding noncommutative space. In this paper, we introduce a kind of quasimartingale spaces which is like but bigger than E(M) and obtain the following interpolation result: let Ê(M) be the space of all bounded E(M)-quasimartingales and 1 < p < pE < qE < q < ∞. Then, there exists a symmetric space F on (0, ∞) with nontrivial Boyd indices such that Ê(M) = (L̂p(M), L̂q(M)F,K.