Noncommutative Davis type decompositions and applications

We prove the noncommutative Davis decomposition for the column Hardy space H^c _p for every (Formula presented.). A new feature of our Davis decomposition is a simultaneous control of (Formula presented.) and (Formula presented.) norms for any noncommutative martingale in (Formula presented.) when (Formula presented.). As applications, we show that the Burkholder/Rosenthal inequality holds for bounded martingales in a noncommutative symmetric space associated with a function space (Formula presented.) that is either an interpolation of the couple (Formula presented.) for some (Formula presented.) or is an interpolation of the couple (Formula presented.) for some (Formula presented.). We also obtain the corresponding (Formula presented.) -moment Burkholder/Rosenthal inequality for Orlicz functions that are either (Formula presented.) -convex and 2-concave for some (Formula presented.) or are 2-convex and (Formula presented.) -concave for some (Formula presented.).