Banach spaces with small Calkin algebras

Abstract

Let $X$ be a Banach space. Let ${\cal A}(X)$ be a closed ideal in the algebra ${\cal L}(X)$ of the operators acting on $X$. We say that ${\cal L}(X)/{\cal A}(X)$ is a Calkin algebra whenever the Fredholm operators on $X$ coincide with the operators