We show that Rochberg's generalizared interpolation spaces $\mathscr Z^{(n)}$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 studysome structural properties of those sequences; in particular, we show thatnontriviality, having strictly singular quotient map, or having strictlycosingular embedding depend only on the basic case $n=k=1$. If we focus on thecase of Hilbert spaces obtained from the interpolation scale of $\ell_p$spaces, then $\mathscr Z^{(2)}$ becomes the well-known Kalton-Peck $Z_2$ space;we then show that $\mathscr Z^{(n)}$ is (or embeds in, or is a quotient of) atwisted Hilbert space only if $n=1,2$, which solves a problem posed by DavidYost; and that it does not contain $\ell_2$ complemented unless $n=1$. Weconstruct another nontrivial twisted sum of $Z_2$ with itself that contains$\ell_2$ complemented.
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机译：我们证明了Rochberg广义插值空间$ \ mathscr Z ^ {（n）} $来自Banach空间的解析族，形成了从$ 0 \ to \ mathscr Z ^ {（n）} \ to \ mathscr Z ^ {（n + k）} \ to \ mathscr Z ^ {（k）} \ to 0 $。我们研究了这些序列的一些结构特性。特别地，我们证明具有平凡的商数图或具有严格的单数嵌入的非平凡性仅取决于基本情况$ n = k = 1 $。如果我们关注从$ \ ell_p $ spaces的插值比例获得的希尔伯特空间的情况，则$ \ mathscr Z ^ {（2）} $成为著名的Kalton-Peck $ Z_2 $空间；然后证明$ \ mathscr Z ^ {{（n）} $仅在$ n = 1,2 $时才是（或嵌入或为）希尔伯特空间，这解决了DavidYost提出的问题；并且除非$ n = 1 $，否则它不包含$ \ ell_2 $补码。我们使用$ \ ell_2 $补余的自身构造$ Z_2 $的另一个平凡的扭曲总和。
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