Calkin algebras for Banach spaces with finitely decomposable quotients

Manuel González ¹   ; José M. Herrera ¹

¹ Departamento de Matemáticas Universidad de Cantabria E-39071 Santander, Spain

Studia Mathematica, Tome 157 (2003) no. 3, pp. 279-293

Résumé

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 $X$.

Détail

DOI : 10.4064/sm157-3-3

Keywords: banach space quotients only admit direct decompositions number summands smaller equal every operator identified times scalar matrix modulo strictly cosingular operators precisely obtain algebra isomorphism calkin algebra subalgebra algebra times scalar matrices which triangularizable indecomposable get information class semi fredholm operators essential spectrum operator acting