Journal of Operator Theory

Volume 53, Issue 2, Spring 2005 pp. 303-314.

Spaces on which the essential spectrum of all the operators is finite

Authors: Manuel Gonz\'alez (1) and Jos\'e M. Herrera (2)
Author institution: (1) Universidad de Cantabria, Departamento de Matem\'aticas, E-39071 Santander, Spain
(2) Universidad de Cantabria, Departamento de Matem\'aticas, E-39071 Santander, Spain

Summary: We study the Banach spaces $X$ for which the essential spectrum $\sigma_\mathrm{e}(T)$ of every $T$ in $L(X)$ is finite. We show that there exists an integer $n$ so that $|\sigma_\mathrm{e}(T)| \leqslant n$ for every $T$. We also show that $X$ admits an irreducible decomposition as a direct sum of indecomposable subspaces, and that the quotient algebra $L(X)/\mathrm{In}(X)$, $\mathrm{In}(X)$ the inessential operators, is isomorphic to a finite product of spaces of scalar matrices.

Keywords: Indecomposable Banach spaces, Fredholm operators, Calkin algebra.

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