Complex interpolation and twisted twisted Hilbert spaces

TL;DR

This paper investigates the structure of interpolation spaces derived from analytic families of Banach spaces, revealing their exact sequences, structural properties, and specific behaviors in the context of Hilbert spaces and twisted Hilbert spaces.

Contribution

It establishes the exact sequence structure of Rochberg's interpolation spaces, analyzes their properties, and characterizes when these spaces relate to twisted Hilbert spaces, solving a problem posed by Yost.

Findings

Exact sequences for interpolation spaces are established.

Nontriviality and embedding properties depend only on the basic case n=k=1.

Only for n=1,2 do the spaces relate to twisted Hilbert spaces.

Abstract

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 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 $\mathscr Z^{(2)}$ 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) 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.