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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.

Calkin algebras for Banach spaces with finitely decomposable quotients