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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.