Complex interpolation and twisted twisted Hilbert spaces

We show that Rochberg’s generalizared interpolation spaces ${\mathcal{Z}}^{(n)}$ arising from analytic families of Banach spaces form exact sequences $0\to \mathscr Z^{(n)} \to \mathscr Z^{(n+k)} \to \mathscr Z^{(k)} \to 0$. 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 ${\mathrm{\ell }}_p$ 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 ${\mathrm{\ell }}_2$ complemented unless $n=1$. We construct another nontrivial twisted sum of $Z_2$ with itself that contains ${\mathrm{\ell }}_2$ complemented.