Spaces on which the essential spectrum of all the operators is finite. (English) Zbl 1228.47018
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.
MSC:
47A53 | (Semi-) Fredholm operators; index theories |
47A10 | Spectrum, resolvent |
46B20 | Geometry and structure of normed linear spaces |
46L80 | \(K\)-theory and operator algebras (including cyclic theory) |