For a Banach space X such that all quotients only admit direct decompositions with a number of summands smaller than or equal to n, we show that every operator T on X can be identified with an n × n scalar matrix modulo the strictly cosingular operators SC(X). More precisely, we obtain an algebra isomorphism from the Calkin algebra L(X)/SC(X) onto a subalgebra of the algebra of n × n scalar matrices which is triangularizable when X is indecomposable. From this fact we get some information on the class of all semi-Fredholm operators on X and on the essential spectrum of an operator acting on X.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-3-3,
author = {Manuel Gonz\'alez and Jos\'e M. Herrera},
title = {Calkin algebras for Banach spaces with finitely decomposable quotients},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {279-293},
zbl = {1032.47003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-3-3}
}
Manuel González; José M. Herrera. Calkin algebras for Banach spaces with finitely decomposable quotients. Studia Mathematica, Tome 157 (2003) pp. 279-293. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-3-3/