Commuting boundedly decomposable perturbations

AbstractThe author proves that perturbation of a decomposable operator by a commuting boundedly decomposable operator is decomposable, thus generalizing a theorem of C. Apostol. It is also proved that the sum of a regular, A-spectral operator and a commuting boundedly decomposable operator is strongly decomposable, while the sum of two commuting boundedly decomposable operators is generalized spectral. Examples show that boundedly decomposable operators form a proper intermediate class between spectral and generalized spectral operators; thus the generalization of Apostol's theorem is not trivial