## Banach spaces with small Calkin algebras

### Volume 75 / 2007

#### 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 whose class in
${\cal L}(X)/{\cal A}(X)$ is invertible.
Among other examples, we have the cases in which ${\cal A}(X)$ is the
ideal of compact, strictly singular, strictly cosingular and
inessential operators, and some other ideals introduced as
perturbation classes in Fredholm theory.
Our aim is to present some classes of Banach spaces and some
concrete examples of Banach spaces for which some of their Calkin
algebras are “small” in some sense: finite dimensional,
commutative, etc. The first example of such a Banach space was
constructed around 1990. However, at this moment there is a great
variety of examples of spaces of this kind, which provides
interesting examples and counterexamples of operators. Moreover, the
methods and techniques of operator theory have been found to be
useful in the study of these spaces.