Complex interpolation and twisted twisted Hilbert spaces

Abstract

We show that Rochberg's generalizared interpolation spaces ${\mathcal{Z}}^{(n)}$ arising from analytic families of Banach spaces form exact sequences $0\to {\mathcal{Z}}^{(n)}\to {\mathcal{Z}}^{(n+k)}\to {\mathcal{Z}}^{(k)}\to 0$. 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 ${\ell }_p$ spaces, then ${\mathcal{Z}}^{(2)}$ becomes the well-known Kalton-Peck $Z_2$ space; we then show that ${\mathcal{Z}}^{(n)}$ is (or embeds in, or is a quotient of) a twisted Hilbert space only if $n=1,2$, which solves a problem posed by David Yost; and that it does not contain ${\ell }_2$ complemented unless $n=1$. We construct another nontrivial twisted sum of $Z_2$ with itself that contains ${\ell }_2$ complemented.

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• Differential processes generated by two interpolators