Extension and lifting of operators and polynomials

TL;DR

This paper investigates the extension and lifting of operators, polynomials, and holomorphic functions within specific operator ideals, providing new characterizations of certain Banach spaces and illustrating differences through examples.

Contribution

It offers new characterizations of  and -spaces, extending prior results with a homological perspective and detailed examples of operator extension and lifting.

Findings

Characterization of  and -spaces using operator ideals

Examples demonstrating differences in extending and lifting operators

Homological approach to understanding extension and lifting problems

Abstract

We study the problem of extension and lifting of operators belonging to certain operator ideals, as well as that of their associated polynomials and holomorphic functions. Our results provide a characterization of $\mathcal{L}_1$ and $\mathcal{L}_{\infty}$-spaces that includes and extends those of Lindenstrauss-Rosenthal \cite{LR} using compact operators and Gonz\'{a}lez-Guti\'{e}rrez \cite{GG} using compact polynomials. We display several examples to show the difference between extending and lifting compact (resp. weakly compact, unconditionally convergent, separable and Rosenthal) operators to operators of the same type. Finally, we show the previous results in a homological perspective, which helps the interested reader to understand the motivations and nature of the results presented.