Complete isomorphic classifications of some spaces of compact operators

Abstract:

This paper is a continuation and a complement of our previous work on isomorphic classification of some spaces of compact operators. We improve the main result concerning extensions of the classical isomorphic classification of the Banach spaces of continuous functions on ordinals. As an application, fixing an ordinal $\alpha$ and denoting by $X^{\xi }$, $\omega _{\alpha } \leq \xi < \omega _{\alpha +1}$, the Banach space of all $X$-valued continuous functions defined in the interval of ordinals $[0, \xi ]$ and equipped with the supremum, we provide complete isomorphic classifications of some Banach spaces ${\mathcal K}(X^{\xi }, Y^\eta )$ of compact operators from $X^\xi$ to $Y^\eta$, $\eta \geq \omega$.

It is relatively consistent with ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that these results include the following cases:

1. $X^{*}$ contains no copy of $c_{0}$ and has the Mazur property, and $Y= c_{0}(J)$ for every set $J$.

2. $X=c_{0}(I)$ and $Y=l_{q}(J)$ for any infinite sets $I$ and $J$ and $1 \leq q < \infty$.

3. $X=l_{p}(I)$ and $Y=l_{q}(J)$ for any infinite sets $I$ and $J$ and $1 \leq q<\!p\! < \infty$.