The Kalton-Peck space as a spreading model

Abstract

The so-called Kalton-Peck space $Z_2$ is a twisted Hilbert space induced, using complex interpolation, by $c_0$ or ${\ell }_p$ for any $1\le p\ne 2<\infty$. Kalton and Peck developed a scheme of results for $Z_2$ showing that it is a very rigid space. For example, every normalized basic sequence in $Z_2$ contains a subsequence which is equivalent to either the Hilbert copy ${\ell }_2$ or the Orlicz space ${\ell }_M$. Recently, new examples of twisted Hilbert spaces, which are induced by asymptotic ${\ell }_p$-spaces, have appeared on the stage. Thus, our aim is to extend the Kalton-Peck theory of $Z_2$ to twisted Hilbert spaces $Z(X)$ induced by asymptotic $c_0$ or ${\ell }_p$-spaces $X$ for $1\le p<\infty$. One of the novelties is to use spreading models to gain information on the isomorphic structure of the subspaces of a twisted Hilbert space. As a sample of our results, the only spreading models of $Z(X)$ are ${\ell }_2$ and ${\ell }_M$, whenever $X$ is as above and $p\ne 2$.

35 pages