COMPLEX INTERPOLATION AND TWISTED TWISTED HILBERT SPACES

We show that Rochberg's generalized interpolation spaces X-(n) arising from analytic families of Banach spaces form exact sequences 0 -> X-(n) -> X(n+k) -> X-(k) -> 0. We study some structural properties of those sequences; in particular, we show that nontriviality, having strictly singular quotient map, or having strictly cosingular embedding depend only on the basic case n = k = 1. If we focus on the case of Hilbert spaces obtained from the interpolation scale of l(p) spaces, then X-(2) becomes the well- known KaltonPeck space Z(2); we then show that X-(n) is (or embeds in, or is a quotient of) a twisted Hilbert space only if n D 1; 2, which solves a problem posed by David Yost; and that it does not contain l(2) complemented unless n = 1. We construct another nontrivial twisted sum of Z(2) with itself that contains l(2) complemented.