Decompositions for real Banach spaces with small spaces of operators

• Autorzy: Manuel González , José M. Herrera
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47A53: (Semi-) Fredholm operators; index theories

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 183
• Numer: 1
• Strony: 1-14

Daty

wydano

2007

Twórcy

autor

Manuel González

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

autor

José M. Herrera

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

Identyfikatory

DOI

10.4064/sm183-1-1