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Calkin algebras for Banach spaces with finitely decomposable quotients

100%

González M., Herrera J.

Studia Mathematica

2003

tom 157

nr 3

279-293

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.

Decompositions for real Banach spaces with small spaces of operators

100%

González M., Herrera J.

Studia Mathematica

2007

tom 183

nr 1

1-14

We consider real Banach spaces X for which the quotient algebra 𝓛(X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_{i}$ for which $𝓛(X_{i})/ℐn(X_{i})$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_{i}$ can be divided into subsets in such a way that if $X_{i}$ and $X_{j}$ are in different subsets, then $𝓛(X_{i},X_{j}) = ℐn(X_{i},X_{j})$; and if they are in the same subset, then $X_{i}$ and $X_{j}$ are isomorphic, up to a finite-dimensional subspace. Moreover, denoting by X̂ the complexification of X, we show that 𝓛(X)/ℐn(X) and 𝓛(X̂)/ℐn(X̂) have the same dimension.

Ograniczanie wyników

2 Studia Mathematica

2 González M.

2 Herrera J.

1 2007

1 2003

Wyniki wyszukiwania

Calkin algebras for Banach spaces with finitely decomposable quotients

Decompositions for real Banach spaces with small spaces of operators