Calkin algebras for Banach spaces with finitely decomposable quotients
Volume 157 / 2003
Studia Mathematica 157 (2003), 279-293
MSC: Primary 47A10, 47A53; Secondary 46B20.
DOI: 10.4064/sm157-3-3
Abstract
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 $n\times n$ 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 $n\times n$ 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 $X$.
