m at h . FA ] 4 D ec 1 99 9 Complemented Subspaces of Spaces Obtained by Interpolation

If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A 0 , A 1) such that A 0 and A 1 are isometric to X ⊕ V , and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon–Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces. There are many Banach space properties that pass to spaces obtained by the complex method of interpolation. For example, it is known that if a couple (A 0 , A 1) is such that A 0 and A 1 both have the UMD (unconditional martingale difference sequence) property, and if A θ is the space obtained using the complex interpolation method with parameter θ, then A θ has the UMD property whenever 0 < θ < 1. Another example is type of Banach spaces: if A 0 has type p 0 and A 1 has type p 1 , then A θ has type p θ , where 1/p θ = (1 − θ)/p 0 + θ/p 1. Similar results are true for the real method of interpolation. If we denote by A θ,p the space obtained using the real interpolation method from a couple (A 0 , A 1) with parameters θ and p, then A θ,p has the UMD property whenever A 0 and A 1 have the UMD property, 0 < θ < 1, and 1 < p < ∞. Similarly, if A 0 has type p 0 and A 1 has type p 1 , then A θ,p has type p θ , where 1/p θ = (1 − θ)/p 0 + θ/p 1 and p = p θ (see [5] 2.g.22). However, there are other properties for which it has been hitherto unknown whether they pass to the intermediate spaces. Examples include the Radon–Nikodym property, the AUMD (analytic unconditional martingale difference sequence) property, and having non-trivial cotype.