#### Vol. 276, No. 2, 2015

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Complex interpolation and twisted twisted Hilbert spaces

### Félix Cabello Sánchez, Jesús M. F. Castillo and Nigel J. Kalton

Vol. 276 (2015), No. 2, 287–307
##### Abstract

We show that Rochberg’s generalized interpolation spaces ${\mathsc{X}}^{\left(n\right)}$ arising from analytic families of Banach spaces form exact sequences $\mathfrak{0}\to {\mathsc{X}}^{\left(n\right)}\to {\mathsc{X}}^{\left(n+k\right)}\to {\mathsc{X}}^{\left(k\right)}\to \mathfrak{0}$. 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 $n=k=\mathfrak{1}$. If we focus on the case of Hilbert spaces obtained from the interpolation scale of ${\ell }_{p}$ spaces, then ${\mathsc{X}}^{\left(\mathfrak{2}\right)}$ becomes the well-known Kalton–Peck space ${Z}_{\mathfrak{2}}$; we then show that ${\mathsc{X}}^{\left(n\right)}$ is (or embeds in, or is a quotient of) a twisted Hilbert space only if $n=\mathfrak{1},\mathfrak{2}$, which solves a problem posed by David Yost; and that it does not contain ${\ell }_{\mathfrak{2}}$ complemented unless $n=\mathfrak{1}$. We construct another nontrivial twisted sum of ${Z}_{\mathfrak{2}}$ with itself that contains ${\ell }_{\mathfrak{2}}$ complemented.

##### Keywords
Complex interpolation, twisted sums of Banach spaces
##### Mathematical Subject Classification 2010
Primary: 46B20, 46B70, 46M18, 46M35